Albeverio S., Fan R., Herzberg F. Hyperfinite Dirichlet Forms And

Lecture Notes of 10 the Unione Matematica Italiana

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The Editorial Policy can be found at the back of the volume. Sergio Albeverio • Ruzong Fan Frederik Herzberg

Hyperﬁnite Dirichlet Forms and Stochastic Processes

123 Sergio Albeverio Frederik Herzberg University of Bonn Bielefeld University Institute for Applied Mathematics and HCM Institute of Mathematical Economics Endenicher Allee 60 Universitätsstraße 25 53115 Bonn 33615 Bielefeld Germany Germany [email protected] [email protected]

Ruzong Fan Texas A and M University Department of Statistics College Station 77843, TX USA [email protected]

Current address Biostatistics and Bioinformatics Branch Division of Epidemiology, Statistics & Prevention Eunice Kennedy Shriver National Institute of Child Health & Human Development 6100 Executive Blvd. MSC 7510, Bethesda, MD 20892 United States of America

ISSN 1862-9113 ISBN 978-3-642-19658-4 e-ISBN 978-3-642-19659-1 DOI 10.1007/978-3-642-19659-1 Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2011928508

Mathematics Subject Classiﬁcation (2000): 03H05; 60J45

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Preface

The theory of stochastic processes has developed rapidly in the past decades. Martingale theory and the study of smooth diffusion processes as solutions of stochastic differential equations have been extended in several directions, such as the study of infinite dimensional diffusion processes, the study of diffusion processes with non-smooth unbounded coefficients, diffusion processes on manifolds and on singular spaces. The interplay between stochastic analysis and mathematical physics has been one of the most important and exciting research areas.

One of the best techniques to deal with the problems of these areas is Dirichlet space theory. In the original framework of this theory, the state space is a locally compact separable metric space, e.g., Rd,orad-dimensional manifold. This theory has given us a nice understanding about the property of diffusion processes with non-smooth unbounded coefficients. Moreover, it has been fruitfully applied to mathematical physics. This framework has been generalized to state spaces which are more general topological spaces or some infinite dimensional vector spaces or manifolds. Several key problems, such as the closability of quadratic forms and the construction of strong Markov processes associated with quasi-regular Dirichlet forms, have been solved. The study of infinite dimensional stochastic analysis as well as the study of processes on singular structures (like fractals, trees, or general metric spaces) has enriched and extended the Dirichlet space theory. In the meantime, a new framework has been introduced into Dirichlet space theory by the development of nonstandard probabilistic analysis [25, 166]. As is well-known, nonstandard analysis is an alternative setting for analysis (and, indeed, all areas of mathematics), namely by enriching the set of real numbers by infinitesimal and infinite elements. It has its origin in seminal work by Schmieden, Laugwitz [325] and most notably Robinson [310]. By now, several textbooks and surveys exist on this theory and its applications (see, e.g. [25, 63, 125, 217]). Nonstandard analysis gives a novel approach to the theory of stochastic processes. In particular, it has led to hyperfinite symmetric Dirichlet space theory. Besides being interesting by itself, it has also many applications. In the first part of the book, we extend the research to the

vii viii Preface nonsymmetric case, and remove some restrictive conditions in the previous treatment of the subject (Chap. 5 of [25]). In addition, we shall apply the theory to present a new approach to inﬁnite dimensional stochastic analysis.

In writing this book we have two main aims: (1) to give a presenta- tion of research on nonsymmetric hyperfinite Dirichlet space theory and its applications in (standard) finite and infinite dimensional stochastic analysis, Chaps. 1–4; (2) to find nonstandard representations for a special class of (finite dimensional) Feller processes and their infinitesimal generators, viz. stochastically continuous processes with stationary and independent increments (i.e., Lévy processes), Chap. 5. Chapter 6 is a complement to illustrate the usefulness of the hyperfinite probability spaces. The first part (Chaps. 1–4) is based on Chap. 5 of Albeverio et al. [25] and the further in depth research of Sergio and Ruzong; the second part (Chaps. 5–6)isbased on results obtained recently by Tom Lindstrøm and their extensions by Sergio and Frederik.

As mentioned earlier, the interplay between stochastic analysis and mathematical physics has been one of the most important and exciting themes of research in the last decades. This is already a sufficient rationale for the research of the first part of the present book. The motivation for including the second part, Chap. 5, into this book is that many of the issues discussed in the more general framework of the first part, such as existence of standard parts of hyperfinite Markov chains, become much less technical to resolve for hyperfinite Lévy processes. Furthermore, the more restrictive setting of the second part also allows one to obtain finer results on the relation between Lévy processes and their hyperfinite analogues, one example being a hyperfinite version of the Lévy–Khintchine formula.

The contents of this book are arranged as follows: In Chap. 1,weintro- duce the framework of hyperfinite Dirichlet forms. We develop the potential theory of hyperfinite Dirichlet forms in Chap. 2. In Chap. 3, we consider standard representations of hyperfinite Markov chains under certain conditions, and translate the conditions on hyperfinite Markov chains into the language of hyperfinite Dirichlet forms. As an interesting and important application in classical stochastic analysis, we construct tight dual strong Markov processes associated with quasi-regular Dirichlet forms by using the language of hyperfinite Dirichlet forms in Chap. 4. The results show that hyperfinite Dirichlet space theory is a powerful tool to study classical problems. In the first sections of Chap. 5, the notion of a hyperfinite Lévy process is introduced and its relation to hyperfinite random walks as well as to standard Lévy processes is investigated. These results can be used to show that the jump part of any Lévy process is essentially a hyperfinite convolution of Poisson processes. Finally, Chap. 6 is an epilogue, providing a rigorous motivation for the study of hyperfinite Loeb path spaces as generic probability spaces. Preface ix

The entire book is based on nonstandard analysis. For the reader’s convenience, we present some basic notions of nonstandard analysis, such as internal sets and saturation, linear spaces, Loeb measure spaces, structure of ∗R and topology in the appendix. Because of its monographical character centered around the hyperﬁnite approach, the book has by no means the goal of including all aspects of recent developments in the theory of stochastic processes and its connections with Dirichlet forms theory or the theory of Lévy processes. For this, we rather refer to surveys and proceedings like Albeverio [2], Barndorﬀ-Nielsen et al. [73], and Ma et al. [275], respectively.

The germ of this book goes back to the year 1989 when the second author, Ruzong Fan, worked on the construction of symmetric Markov processes associated with Dirichlet forms at Peking University, Beijing ([165]andChap.4). At that time, Ruzong was unaware that Sergio’s group was working on the same project using standard methods [41]. The second author, Dr. Zhiming Ma, of [41] did privately inquire Ruzong about the progress of Ruzong’s research in 1989 at the Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing. In response to Dr. Ma’s request of a private meeting, Ruzong presented his work to Dr. MainaclassroomwithDr.Maasthe only audience. Dr. Ma, however, did not mention his ongoing work with Sergio in any way. Thus, Ruzong was totally unaware of Sergio’s research. In the spring of 1990, Ruzong first realized this when he saw a manuscript of Albeverio and Ma [41] in Beijing with a surprise. These events notwith- standing, Ruzong continued to work on a “symmetric version” of Chaps. 1–4 using non-standard language when he was at Peking University till 1991 and when he visited the Humboldt-University, Berlin, between 1991 and 1992. Under Sergio’s supervision and encouragement, Ruzong extended the project to the current “nonsymmetric version” from 1992 to 1994 at Ruhr-University, Bochum. In 2006, Frederik kindly joined the project with a contribution on hyperfinite Lévy processes (Chap. 5) and the Epilogue (Chap. 6). In the summer of 2006, the three authors gathered at the University of Bonn to finalize this monograph. We gratefully acknowledge the manifold support of various institutions in the long process of work on this project.

In the run-up to its completion, Sergio and Frederik were supported par- tially by the collaborative research center SFB 611 of the German Research Foundation (DFG), Germany; in addition, Ruzong’s visit to Bonn was par- tially funded through a research fellowship from the Alexander von Humboldt Foundation, Germany.

Over the course of his career, Ruzong has received a lot of generous support from Sergio. As a Ph.D candidate in Beijing around 1987–1988, Ruzong was greatly fascinated by Sergio and Raphael Høegh-Krohn’s novel work on inﬁnite dimensional stochastic analysis, in which Ruzong ﬁnished his Ph.D thesis. Unfortunately, Ruzong got no chance to meet Raphael Høegh-Krohn; right before Ruzong went to Europe, he was shocked to learn that Raphael x Preface

Høegh-Krohn died of a heart attack. In a relatively isolated environment, Ruzong mostly worked on himself by reading numerous papers and books of Sergio and Raphael Høegh-Krohn; and many times, Ruzong had to spend a few days on a single equation or lemma to guess and to understand it. Whilst it seemed like a helpless or hopeless situation for Ruzong at that time, Ruzong eventually came to the forefront of research in areas of infinite dimensional stochastic analysis: he studied the hard and central questions regarding Beurling–Deny formulae, representation of martingale additive functionals and absolute continuity of symmetric diffusion processes on Banach spaces, potential theory of symmetric hyperfinite Dirichlet forms, and construction of the symmetric strong Markov processes associated with quasi-regular Dirich- let forms by using the non-standard analysis language. This direction of research was initiated by Sergio, although Ruzong was unaware that Sergio’s group already worked on the construction of Markov processes using the language of standard stochastic analysis.

In early 1989, Ruzong applied for a fellowship from the Alexander von Humboldt Foundation from Peking University, Beijing; soon after a rejection from the Foundation in the fall 1989, Ruzong received a warm letter from Sergio with encouragement and a kind oﬀer to nominate, as an academic host, Ruzong for the fellowship and by writing a strong letter of recommendation. This is just one anecdote to illustrate how Ruzong has constantly been able to count on Sergio’s help via communications by either mail or face-to-face conversations starting from 1989. Between 1992 and 1994, Sergio generously supported Ruzong at Ruhr-University Bochum to complete the main part of Chaps. 1–4 of this monograph, and helped Ruzong to pass the hard period of time in his career.

The story of Ruzong is an example how Sergio has helped many young mathematicians to grow and to mature. Quite probably, Ruzong would have disappeared from academia a long time ago without the support of Sergio. In a true sense, Sergio has been an academic father ﬁgure for Ruzong when he des- perately needed one. In recent years, after his departure from Sergio’s research group, Ruzong has been mainly working on statistical genetics guided by his beloved American mentor, Dr. Kenneth Lange, at the University of Michigan and UCLA. Nevertheless, Ruzong has fond memories and deep appreciation of numerous communications with his European academic father Sergio; and both Ruzong and Frederik are deeply grateful for Sergio’s mentoring.

Thus, especially right after Sergio’s 70th birthday in 2009 – which also marks the 50th anniversary of his remarkable scientiﬁc career –, Ruzong and Frederik are sure that they will be joined by many other young mathematicians in thanking Sergio for his wonderful role in our professional and personal development and in wishing him all the best for the rest of his life: Not just continued productivity, but most of all good health, happiness, joy, and peace. Preface xi

We owe a huge debt of gratitude to our families: In the summer of 2006, Dr. Li Zhu (Ruzong’s wife) kindly took care of two young children when her husband was visiting Bonn. Their adorable daughter, Olivia Wenlu Fan, was with the second author in Germany for the “hot and interesting” summer of Bonn, where she liked everything except German milk. Frederik thanks his wife, Angélique Herzberg, for her love and manifold support with the words of Proverbs 31,10–12: “A wife of noble character [...] isworthfarmorethan rubies.Herhusband[...]lacksnothingofvalue.Shebringshimgood [...] all the days of her life.” We are all very grateful to our families for their love and understanding during the entire process of writing this book.

Finally, we would like to thank Dr. Catriona Byrne as well as Susanne Denskus and Ute McCrory of Springer Verlag for their kind, unfainting editorial assistance in the long process of publishing this work.

Contents

1 Hyperfinite Dirichlet Forms ...... 1 1.1 Hyperfinite Quadratic Forms ...... 2 1.2 Domain of the Symmetric Part ...... 6 1.3 Resolvent of the Symmetric Part ...... 17 1.4 Weak Coercive Quadratic Forms...... 26 1.5 Hyperfinite Dirichlet Forms ...... 36 1.6 Hyperfinite Representations ...... 52 1.7 Weak Coercive Quadratic Forms, Revisited ...... 61

2 Potential Theory ...... 65 2.1 Exceptional Sets...... 66 2.1.1 Exceptional Sets ...... 66 2.1.2 Co-Exceptional Sets ...... 71 2.2 Excessive Functions and Equilibrium Potentials ...... 73 2.3 Capacity Theory ...... 78 2.4 Relation of Exceptionality and Capacity Theory ...... 86 2.5 Measures of Hyperﬁnite Energy Integrals ...... 91 2.6 Internal Additive Functionals and Associated Measures ...... 102 2.7 Fukushima’s Decomposition Theorem ...... 107 2.7.1 Decomposition Under the Individual Probability Measures Pi ...... 107 2.7.2 Decomposition Under the Whole Measure P ...... 118 2.8 Internal Multiplicative Functionals ...... 120 2.8.1 Internal multiplicative functionals ...... 120 2.8.2 Subordinate Semigroups ...... 121 2.8.3 Subprocesses ...... 122 2.8.4 Feynman-Kac Formulae...... 123 2.9 Alternative Expression of Hyperﬁnite Dirichlet Forms ...... 124 2.10 Transformations of Symmetric Dirichlet Forms ...... 125

xiii xiv Contents

3 Standard Representation Theory ...... 129 3.1 Standard Parts of Hyperfinite Markov Chains ...... 130 3.1.1 Inner Standard Part of Sets ...... 131 3.1.2 Strong Markov Processes and Modified Standard Parts ...... 136 3.2 Hyperfinite Dirichlet Forms and Markov Processes ...... 145 3.2.1 Separation of Points...... 146 3.2.2 Nearstandardly Concentrated Forms ...... 151 3.2.3 Quasi-Continuity ...... 155 3.2.4 Construction of Strong Markov Processes ...... 160

4 Construction of Markov Processes ...... 165 4.1 Main Result ...... 166 4.2 Hyperﬁnite Lifts of Quasi-Regular Dirichlet Forms ...... 170 4.3 Relation with Capacities...... 177 4.4 Path Regularity of Hyperﬁnite Markov Chains ...... 180 4.5 Quasi-Continuity and Nearstandard Concentration ...... 181 4.6 Construction of Strong Markov Processes...... 192 4.7 Necessity for Existence of Dual Tight Markov Processes ...... 197

5 Hyperfinite Lévy Processes and Applications ...... 199 5.1 Standard Lévy Processes ...... 200 5.2 Characterizing Hyperfinite Lévy Processes...... 203 5.3 Hyperfinite Lévy Processes: Standard Parts ...... 217 5.4 Hyperfinite Lévy-Khintchine Formula ...... 224 5.5 Representation Theorem for Lévy Processes ...... 232 5.6 Extensions and Applications ...... 239

6 Genericity of Loeb Path Spaces...... 243 6.1 Adapted Probability Logic ...... 244 6.2 Universality, Saturation, Homogeneity ...... 245 6.3 Hyperﬁnite Adapted Spaces ...... 246 6.4 Probability Logic and Markov Processes ...... 247

Appendix ...... 249 A.1 General Topology ...... 249 A.2 Structure of ∗R ...... 250 A.3 Internal Sets and Saturation ...... 251 A.4 Loeb Measure...... 252 A.5 Linear Spaces ...... 253

Historical Notes...... 255

References...... 263

Notation Index ...... 281

Index ...... 283 Chapter 1 Hyperﬁnite Dirichlet Forms

The interplay between methods from functional analysis and the theory of stochastic processes is one of the most important and exciting aspects of mathematical physics today. It is a highly technical and sophisticated theory based on decades of research in both areas. Numerous papers have been written on the standard theory of Dirichlet forms. Apart from the articles and monographs cited below, other notable contributions to the area include: Albeverio and Bernabei [5], Albeverio, Kondratiev, and Röckner [32], Albev- erio and Kondratiev [33], Albeverio and Ma [39], Albeverio, Rüdiger, and Wu [54], Bliedtner [94], Bouleau [98], Bouleau and Hirsch [99], Chen et al. [112], Chen, Ma, and Röckner [116], Eberle [149], Exner [154], Fabes, Fukushima, Gross, Kenig, Röckner, and Stroock [155], Fitzsimmons and Kuwae [172], Fukushima [177,179,180], Fukushima and Tanaka [185], Fukushima and Ying [188, 189], Gesztesy et al. [191, 192], Grothaus et al. [198], Hesse et al. [208], Jacob [218–220], Jacob and Moroz [221], Jacob and Schilling [222], Jost et al. [225], Kassmann [232], Kim et al. [240], Kumagai and Sturm [248], Le Jan [258], Liskevich and Röckner [265], Ma and Röckner [272, 273], Ma et al. [274], Mosco [283], Okura [292], Oshima [294, 295], D.W. Robinson [312], Röckner and Wang [317], Röckner and Zhang [319], Schmuland and Sun [329], Shiozawa and Takeda [331], da Silva et al. [332], Stannat [336, 338], Stroock [340], Sturm [343], Takeda [346, 347], Wu [363], and Yosida [364].

In this monograph, we present the theory of Dirichlet forms from a uniﬁed vantage point, using nonstandard analysis, thus viewing the continuum of the time line as a discrete lattice of inﬁnitesimal spacing. This approach is close in spirit to the discrete classical formulation of Dirichlet space theory in A. Beurling and J. Deny’s seminal article [87].

The discrete setup in this monograph permits to study the diffusion and the jump part by essentially the same methods. This setting being independent of special topological properties of the state space, it is also considerably less technical than other approaches. Thus, the theory has found its natural setting and no longer depends on choosing particular topological spaces; in particular, it is valid for both finite and infinite dimensional spaces.

S. Albeverio et al., Hyperﬁnite Dirichlet Forms and Stochastic Processes, 1 Lecture Notes of the Unione Matematica Italiana 10, DOI 10.1007/978-3-642-19659-1_1, c Springer-Verlag Berlin Heidelberg 2011 2 1 Hyperﬁnite Dirichlet Forms

Whilst Albeverio et al. [25], Chap. 5, only discussed symmetric hyperﬁnite Dirichlet forms and related Markov chains (refer to [165, 166]also),weshall extend the theory to the nonsymmetric case. We shall try to follow as much as possible the path suggested by the work on the symmetric case.

An important sub-class of Markov process are Feller processes with stationary and independent increments (Lévy processes), and in recent years, these processes have attracted a lot of interest, including from nonstandard analysts. Initiated by T. Lindstrøm [263], a number of articles have been devoted to the investigation of hyperfinite Lévy processes. Chapter 5 of this monograph is a detailed exposition of Lindstrøm’s theory [263] and its subsequent continuation by Albeverio and Herzberg [14]. The book ends with an expository summary (without proofs) of the model theory of stochastic processes as developed by H.J. Keisler and his coauthors, who formulated and proved the “universality” of hyperfinite adapted probability spaces in a rigorous manner, and a short description of recent fundamental results about the definability of nonstandard universes.

Meanwhile, our purpose in the first chapter is to develop a general theory of hyperfinite quadratic forms. We shall set the scene in Sect. 1.1. Sections 1.2 and 1.3 will study the domains of symmetric parts, the standard parts and resolvents. We shall discuss the property of weak coercive quadratic forms in Sects. 1.4 and 1.7. In Sect. 1.5, we shall study Markov forms and begin the analysis of associated Markov chains and get the basic Beurling–Deny formula. We discuss the hyperfinite lifting theory of standard Dirichlet forms in Sect. 1.6.

1.1 Hyperﬁnite Quadratic Forms

We shall develop a hyperﬁnite theory of nonnegative quadratic forms on inﬁnite dimensional spaces. It is well-known that in the Hilbert space case the theory of closed forms of this kind is equivalent to the theory of nonnegative operators. In fact, there is a natural correspondence between forms E(·, ·) and operators A given by E(u, u)=Au, u,where·, · is the scalar product in the Hilbert space. We have chosen to present the theory in terms of forms and not operators for two reasons: partly because forms are real-valued, and this makes it simpler to take standard parts, but also because in most of our applications, the form is what is naturally given.

Let H be an internal, hyperﬁnite dimensional linear space1 equipped with an inner product ·, · generating a norm ||·||.Let∗R be the nonstandard

1 The notions of hyperﬁnite dimensional linear space are given in Albeverio et al. [25]. 1.1 Hyperﬁnite Quadratic Forms 3 real line2.WecallamapE : H × H −→ ∗R nonnegative quadratic form if and only if for all α ∈ ∗R,u,v,w∈ H,

E(u, u) ≥ 0, E(αu, v)=αE(u, v), E(u, αv)=αE(u, v), E(u + v, w)=E(u, w)+E(v, w), E(w, u + v)=E(w, u)+E(w, v).

Since E(·, ·) is a nonnegative quadratic form on the hyperﬁnite dimensional space H, elementary linear algebra tells us that there is a unique nonnegative deﬁnite operator A : H −→ H such that

E(u, v)=Au, v for all u, v ∈ H. (1.1.1)

∗N 3 { | 1 ≤ ≤ } To see this, let 0 be the nonstandard integers .Let ei i N be an ∗ N orthonormal basis of (H, ·, ·) for an N ∈ N. We put Aei = j=1 E(ei,ej)ej. Then (1.1.1) follows immediately. Hence, A is given by the matrix A = (E(ei,ej))1≤i,j≤N , i.e., ⎛ ⎞ E(e1,e1) E(e1,e2) ... E(e1,eN ) ⎜ ⎟ ⎜ E(e2,e1) E(e2,e2) ... E(e2,eN ) ⎟ A = ⎜ . . . . ⎟ . (1.1.2) ⎝ . . .. . ⎠ E(eN ,e1) E(eN ,e2) ... E(eN ,eN )

Moreover, Au, u≥0 for all u ∈ H. This means that A is a hyperﬁnite dimensional matrix (not necessarily symmetric). Let Aˆ be the adjoint operator of A,thatis,

E(u, v)=u, Avˆ for all u, v ∈ H.

By (1.1.2), we have that Aˆ is the transpose of A.If||A|| and ||Aˆ|| are the operator norms of A and Aˆ, respectively, we have ||A|| = ||Aˆ||.Weﬁxan inﬁnitesimal4 Δt such that

2 ∗R is the standard notation for the nonstandard real line, refer to Appendix, Albeverio et al. [25], Cutland [125], Davis [135], Hurd [216],HurdandLoeb[217], Lindstrøm [262], Stroyan and Bayod [341], and Stroyan and Luxemburg [342]. 3 ∗ N0 is the standard notation for the nonstandard integers, refer to Appendix, Albeverio et al. [25], Cutland [125], Davis [135], Hurd [216],HurdandLoeb[217], Lindstrøm [262], Stroyan and Bayod [341], and Stroyan and Luxemburg [342]. 4 In the sense of nonstandard analysis, refer to Appendix, Albeverio et al. [25], Keisler [237, 238], Stroyan and Bayod [341], and Stroyan and Luxemburg [342]. 4 1 Hyperﬁnite Dirichlet Forms 1 1 0 <Δt≤ = . (1.1.3) ||A|| ||Aˆ||

Let us deﬁne new operators QΔt and QˆΔt by

QΔt = I − ΔtA, QˆΔt = I − ΔtA.ˆ

The relation (1.1.3) implies that the operators QΔt and QˆΔt are nonnegative. Because A is nonnegative, the operator norms of QΔt and QˆΔt are less than or equal to one. Similarly, we deﬁne the nonnegative quadratic co-form Eˆ(·, ·) of E(·, ·) by

Eˆ(u, v)=E(v, u) for all u, v ∈ H.

Introduce a nonstandard time line T by

∗ T = {kΔt | k ∈ N0}.

For each element t = kΔt in T , deﬁne Qt and Qˆt to be the operators

Qt =(QΔt)k, Qˆt =(QˆΔt)k.

t ˆt The families {Q }t∈T and {Q }t∈T are obviously semigroups. We shall call t ˆt {Q }t∈T the semigroup and {Q }t∈T the co-semigroup associated with E(·, ·) and Δt, respectively. Whenever we refer to E(·, ·), Eˆ(·, ·),A,A,ˆ T, Qt and Qˆt in the rest of this book, we shall assume that they are linked by above relations.

t In applications, the primary objects will often be the semigroup {Q }t∈T ˆt ˆ and co-semigroup {Q }t∈T . We can then deﬁne A and A (and hence E(·, ·))by 1 A = I − QΔt , Δt 1 Aˆ = I − QˆΔt . Δt

The operator A is called the infinitesimal generator of E(·, ·), and Aˆ is called the infinitesimal co-generator of E(·, ·).Foreacht ∈ T, we may define approximations A(t) of A and Aˆ(t) of Aˆ by 1 A(t) = I − Qt , t 1 Aˆ(t) = I − Qˆt . (1.1.4) t 1.1 Hyperfinite Quadratic Forms 5

From A(t) and Aˆ(t),wegettheforms

E(t)(u, v)=A(t)u, v = u, Aˆ(t)v, (1.1.5) and

Eˆ(t)(u, v)=E(t)(v, u) = Aˆ(t)u, v = A(t)v, u.

We deﬁne the symmetric part E(·, ·) and anti-symmetric part E˚(·, ·) of E(·, ·) by

1 E( )= E( )+E( ) u, v 2 u, v v, u , 1 E˚( )= E( ) −E( ) u, v 2 u, v v, u .

For α ∈ ∗R,α≥ 0, we set

Eα(u, v)=E(u, v)+αu, v.

Each of these forms generates a norm (possibly a semi-norm in the case α =0):

|u|α = Eα(u, u) = Eα(u, u).

We recall that the original Hilbert space norm on H is denoted by || · ||. Similarly, we set for α ∈ ∗R,α≥ 0,

Eα(u, v)=E(u, v)+αu, v, ˆ ˆ Eα(u, v)=E(u, v)+αu, v.

t Let A and {Q } be the generator and semigroup of E(·, ·), respectively. Then 1 1 = + ˆ Δt = Δt + ˆΔt kΔt =( Δt)k ∀ ∈ ∗N A 2 A A , Q 2 Q Q and Q Q , k .

t Since A and Q are nonnegative, self-adjoint operators, they have unique 1 t nonnegative square roots, which we denote by A 2 and Q 2 , respectively. 6 1 Hyperﬁnite Dirichlet Forms

In the same manner as (1.1.4)and(1.1.5), we can deﬁne approximations (t) (t) A of A and E (·, ·) of E(·, ·) by

(t) 1 t (t) (t) A = I − Q , E (u, v)=A u, v,t∈ T. t

If a nonnegative quadratic form E(·, ·):H × H −→ ∗R satisﬁes

E(u, v)=E(v, u) for all u, v ∈ H, i.e., E(u, v)=E(u, v), we shall call it a nonnegative symmetric quadratic form. It is easy to see that a nonnegative quadratic form E(u, v) is symmetric if and only if A = Aˆ or Qt = Qˆt, ∀t ∈ T.

In this book, we shall deal with nonnegative quadratic forms E(·, ·) and the related theory. For the framework, potential theory and applications of nonnegative symmetric quadratic form, we refer the reader to Albeverio et al. [25], Chap. 5, Sect. 5.1 and Fan [165, 166]. We shall utilize the known results of symmetric forms in our study, and extend them to the nonsymmetric case. In particular, we need the notion of the symmetric part E(·, ·) of E(·, ·), and the related notations. In Sect. 1.2, we shall define the domain D(E) of the t symmetric part E(·, ·) by using the semigroup {Q | t ∈ T }. We shall introduce ∗ the resolvent {Gα | α ∈ (−∞, 0)} of E(·, ·) in Sect. 1.3, and characterize the domain D(E) by this resolvent. In Sect. 1.4, we shall define the domain D(E) of ∗ E(·, ·) by its resolvent {Gα | α ∈ (−∞, 0)}; under the hyperfinite weak sector condition, we shall show that D(E)=D(E). In Sect. 1.5, we shall introduce hyperfinite Dirichlet forms and related Markov chains. For standard coercive forms, we shall construct their nonstandard representation in Sect. 1.6.

1.2 Domain of the Symmetric Part

In this section, we shall define the domain D(E) of the symmetric part E(·, ·) for a hyperfinite nonnegative quadratic form E(·, ·). Before giving a strict definition (Definition 1.2.1), we shall mention an intuitive description. At first, let Fin(H) be the set of all elements in H with finite norm. By defining x ≈ y if ||x − y|| ≈ 0, we know from Proposition A.5.2 in the Appendix that the space5

◦H =Fin(H)/ ≈

5 ≈ stands for diﬀering by an inﬁnitesimal, in the sense of nonstandard analysis, refer to Albeverio et al. [25], Cutland [125], Davis [135], Hurd [216], Hurd and Loeb [217], and Lindstrøm [262]. 1.2 Domain of the Symmetric Part 7 is a Hilbert space with respect to the inner product (◦x, ◦y)=st(x, y), where ◦x denotes the equivalence class of x and st : ∗R −→ R is the mapping of standard part6.Wecall(◦H,(·, ·)) the hull of (H, ·, ·).

Consider the standard part E(·, ·) of the nonnegative symmetric quadratic form E(·, ·).IfE(·, ·) is S-bounded, i.e., there exists a constant K ∈ R+ such that

|E(u, v)|≤K||u||||v|| for all u, v ∈ H, we can simply deﬁne E(·, ·) by

E(◦u, ◦v)=◦E(u, v).

If E(·, ·) is not S-bounded, we shall meet two difficulties. We no longer have that E(u, v) ≈ E(˜u, v˜) whenever u ≈ u˜ and v ≈ v˜, and there may be elements v ∈ Fin(H) such that E(˜v,v˜) is infinite for all v˜ ≈ v.Thelatter problem should not surprise us. It is an immediate consequence of the fact that unbounded forms on Hilbert spaces cannot be defined everywhere. We shall solve it by simply letting E(◦u, ◦v) be undefined when E(˜v,v˜) is infinite for all v˜ ∈ ◦v. The most natural solution to the first problem may be to define

E(◦u, ◦u)=inf{◦E(v, v) | v ∈ ◦u}, (1.2.1) andthenextendE(·, ·) to be a bilinear form by the usual trick 1 (◦ ◦ )= (◦ + ◦ ◦ + ◦ ) − (◦ ◦ ) − (◦ ◦ ) E u, v 2 E u v, u v E u, u E v, v .

The disadvantage of this approach is that it gives us little understanding of how the inﬁmum in (1.2.1) is obtained. For an easier access to the regularity properties of E(·, ·) and E(·, ·), we prefer a more indirect way of attack. Our plan is to deﬁne a subset D(E) of Fin(H) – we call it the domain of E(·, ·) – satisfying

if ◦E(u, u) < ∞, there is a v ∈D(E) such that v ≈ u, (1.2.2) if u, v ∈D(E) and u ≈ v, then ◦E(u, u)=◦E(v, v) < ∞. (1.2.3)

We then deﬁne E(·, ·) by

E(◦u, ◦u)=◦E(v, v), (1.2.4)

6 Refer to Albeverio et al. [25]. 8 1 Hyperﬁnite Dirichlet Forms when v ∈D(E) ∩ ◦u. It turns out that the two deﬁnitions (1.2.1)and(1.2.4) agree (see Proposition 1.2.4).

If we look at the standard nonsymmetric Dirichlet theory, see Albeverio et al. [9], Kim [241] and Ma and Röckner [270], the domain of a quadratic form is given from the very beginning. After that, the authors such as those of Ma and Röckner [270] introduced the symmetric and anti-symmetric parts (see page 15, [270]). This method makes the domains of the quadratic form and its symmetric part coincide. On the other hand, Albeverio et al. [25] has given us a very nice definition of domain for the symmetric hyperfinite quadratic forms by their semigroups. Therefore, we may define the domain t D(E) of E(·, ·) via the semigroup of {Q | t ∈ T }. In the next section, we shall ∗ discuss the property of the resolvent {Gα | α ∈ (−∞, 0)} of E(·, ·).Wecan ∗ define the domain of D(E) through {Gα | α ∈ (−∞, 0)}.

Now it is very natural to ask: can we as well define the domain D(E) of E(·, ·) directly from {Qt | t ∈ T }? Here we would mention that it seems not easy to do the job. In Sect. 1.4, we shall define D(E) by means of the resolvent {Gα | α<0} of E(·, ·). Under the hypothesis of weak sector condition, we shall prove D(E)=D(E) by showing that the two definitions satisfy (1.2.1). This is similar to the procedure in the standard nonsymmetric Dirichlet space theory, see, e.g., Albeverio et al. [9], Albeverio et al. [47], Albeverio and Ugolini [57], Kim [241], and Ma and Röckner [270].

(t) Notice that even when E(·, ·) is not S-bounded, E (·, ·) is S-bounded for all non-inﬁnitesimal t. One of the motivations behind our deﬁnition of the domain D(E) is that we want to single out the elements where E(·, ·) is really (t) approximated by the bounded forms E (·, ·),t ≈ 0, i.e., those u ∈ H such that

(t) ◦E(u, u) = lim ◦E (u, u). (1.2.5) t↓0 t≈0

We could have taken this to be our deﬁnition of D(E), but for technical and expository reasons we have chosen another one which we shall soon show to be equivalent to (1.2.5) (see Proposition 1.2.2).

Deﬁnition 1.2.1. Let E(·, ·) be a nonnegative quadratic form on a hyperﬁ- nite dimensional linear space H. The domain D(E) of the symmetric part of E(·, ·) is the set of all u ∈ H satisfying ◦ ◦ (i) E1(u, u)= E1(u, u) < ∞. t t (ii) For all t ≈ 0, E(Q u, Q u) ≈ E(u, u).

Let us try to convey the intuition behind this deﬁnition. Thinking of A as a diﬀerential operator, the elements of D(E) are “smooth” functions and 1.2 Domain of the Symmetric Part 9

t Q is a “smoothing” operator often given by an integral kernel. If an element t u is already smooth, then an inﬁnitesimal amount of smoothing Q ,t ≈ 0, t t should not change it noticeably, and hence E(Q u, Q u) ≈ E(u, u). We shall give a partial justiﬁcation of this rather crude image later, when we show ◦ t that if E1(u, u) < ∞, then the “smoothed” elements Q u, t ≈ 0, are all in D(E) (Lemma 1.2.3, see also Corollary 1.2.3).

Our ﬁrst task will be to establish a list of alternative deﬁnitions of D(E), among them (1.2.5). We begin with the following simple identity giving the relationship between E(·, ·) and E(t)(·, ·), and also the relationship between (t) E(·, ·) and E (·, ·):

Lemma 1.2.1. For all u ∈ H and t ∈ T , we have

(t) (i) E(t)(u, u) ≥ 0 and E (u, u) ≥ 0, Δt Δt (ii) E(t)(u, u)= E(Qsu, u)= E(u, Qˆsu), t 0≤s

Proof. (i) We have 1 E(t)(u, u)= (I − Qt)u, u t 1 = u, u−Qtu, u t 1 ≥ u, u−||Qt||u, u t ≥ 0,

t Δt t (t) since ||Q || ≤ ||Q || Δt ≤ 1. It is then easy to see that E (u, u) ≥ 0.

(ii) By an easy calculation, we have 1 E(t)(u, u)= (I − Qt)u, u t 1 = (Qs − Qs+Δt)u, u t 0≤s

(iii) In the same way as (1.2.6), we can prove that the ﬁrst equation holds. The second one is due to the symmetry and the semigroup property s of Q .

(t) Among other things, Lemma 1.2.1 tells us that E(t)(·, ·) and E (·, ·) are nonnegative.

Lemma 1.2.2. Let B,C : H −→ H be nonnegative, symmetric operators commuting with A and each other. Then the functions

t (s) t t →Q Bu,Cu and t → E (Q Bu,Cu)

are nonnegative and decreasing for all u ∈ H and s ∈ T .

(s) Proof. We ﬁrst notice that the E (·, ·) part follows from the other one since

(s) t 1 t s E (Q Bu,Cu)= Q (I − Q )Bu,Cu, s s and the operator B =(I − Q )B is nonnegative and commutes with A and C. If t>r,then

r t t−r r Q Bu,Cu−Q Bu,Cu = (I − Q )Q Bu,Cu (t−r) r/2 r/2 =(t − r)E (Q B1/2C1/2u, Q B1/2C1/2u) ≥ 0,

(t−r) t whereweusedthatE (·, ·) is nonnegative. Hence, t −→ Q Bu,Cu decreases. For the positivity, we observe that

t t/2 t/2 Q Bu,Cu = Q B1/2C1/2u, Q B1/2C1/2u ≥ 0.

From Lemma 1.2.2 we may now obtain our main inequalities.

Proposition 1.2.1. For all u ∈ H, t ∈ T : t t t t (i) 0 ≤ E(u, u − Q u) ≤ E(u, u) − E(Q u, Q u) ≤ 2E(u, u − Q u).

Δt Δt 2Δt 2Δt Δt Δt (ii) 0 ≤ E(Q u, Q u) − E(Q u, Q u) ≤ E(u, u) − E(Q u, Q u).

Proof. By trivial algebra, we have

t t t t t E(u, u) − E(Q u, Q u)=E(u, u − Q u)+E(Q u, u − Q u). 1.2 Domain of the Symmetric Part 11

t Applying Lemma 1.2.2 with B = I,C = I − Q , we see that

t t t 0 ≤ E(Q u, u − Q u) ≤ E(u, u − Q u), and part (i) follows.

(ii) The non-negativity is immediate from (i), and as above we have

Δt Δt Δt Δt Δt E(u, u) − E(Q u, Q u)=E(u, u − Q u)+E(Q u, u − Q u).

Applying Lemma 1.2.2 to each of the latter two terms, using B = I,C = Δt Δt Δt I − Q in the ﬁrst case, and B = Q ,C = I − Q in the second, we get

Δt Δt 2Δt Δt 3Δt Δt E(u, u) − E(Q u, Q u) ≥ E(Q u, u − Q u)+E(Q u, u − Q u) 2Δt 2Δt Δt = E(Q u, u) − E(Q u, Q u) 3Δt 3Δt Δt +E(Q u, u) − E(Q u, Q u) Δt Δt 2Δt 2Δt = E(Q u, Q u) − E(Q u, Q u).

The proposition is proved.

The inequalities above are what we need to establish a reasonable characterization of D(E). We ﬁrst give our promised list of alternative deﬁnitions of the domain of E(·, ·).

Proposition 1.2.2. The following statements are equivalent: (i) u is in the domain D(E) of E(·, ·). ◦ ◦ t (ii) E1(u, u)= E1(u, u) < ∞, and for all t ≈ 0, we have E(u, u−Q u) ≈ 0. ◦ t t (iii) E1(u, u) < ∞, and for all t ≈ 0, we have E(u − Q u, u − Q u) ≈ 0. ◦ (t) (iv) E1(u, u) < ∞,andforallt ≈ 0, we have E (u, u) ≈ E(u, u).

Proof. (i) ⇐⇒ (ii). Follows immediately from Proposition 1.2.1 (i).

(ii) =⇒ (iii).Wehave

t t t t t 0 ≤ E(u − Q u, u − Q u)=E(u, u − Q u) − E(Q u, u − Q u),

t t and by Lemma 1.2.2 the term E(Q u, u − Q u) is positive.

(iii) =⇒ (i). We recall that |u|0 = E(u, u) is a semi-norm. By Lemma 1.2.2 and the triangle inequality, we have

t t 0 ≤|u|0 −|Q u|0 ≤|u − Q u|0. 12 1 Hyperﬁnite Dirichlet Forms

t Multiplying both sides by |u|0 + |Q u|0, we get

2 t 2 t t t 0 ≤|u|0 −|Q u|0 ≤|u − Q u|0(|u|0 + |Q u|0) ≤ 2|u|0|u − Q u|0.

t t Hence if ◦E(u, u) < ∞ and E(u − Q u, u − Q u) ≈ 0, we have that

t t E(u, u) − E(Q u, Q u) ≈ 0.

(ii) =⇒ (iv). Follows at once from Lemma 1.2.1. s (iv) =⇒ (ii). Follows from Lemma 1.2.1 and the fact that s → E(Q u, u) is decreasing.

The characterizations of D(E) given in the Proposition 1.2.2 are useful for diﬀerent purposes. As an illustration, we use Proposition 1.2.2 (iii) to prove that the domain has the right linear structure.

Corollary 1.2.1. Let u, v ∈D(E), and assume that α ∈ ∗R is a nearstan- dard7 number. Then αu and u + v are elements of D(E).

Proof. The αu part is trivial. For u + v we use Proposition 1.2.2 (iii) and the triangle inequality.

t t t ||(u + v) − Q (u + v)||0 = ||u − Q u + v − Q v||0 t t ≤||u − Q u||0 + ||v − Q v||0.

The latter two terms above are inﬁnitesimals when t ≈ 0. (δ) Corollary 1.2.2. For any inﬁnitesimal δ ∈ T , we have D(E) ⊂D(E ).

(kδ) Proof. Let u ∈D(E).ByProposition1.2.2 (iv), we know E (u, u) ≈ (δ) E(u, u) ≈ E (u, u) for all k such that kδ ≈ 0.ByProposition1.2.2 (iv) (δ) again, we get u ∈D(E ).

t The second part of Proposition 1.2.1 informsusthatQ u is more likely to be in D(E) than u is. The next lemma pins this down more precisely.

◦ Lemma 1.2.3. Assume E1(u, u) < ∞. Then for all non-inﬁnitesimals t,we t have Q u ∈D(E).

Proof. By Proposition 1.2.1,wehave

◦ t t ◦ E 1(Q u, Q u) ≤ E1(u, u) < ∞.

7 See Appendix and Albeverio et al. [25] for the concept of nearstandard. 1.2 Domain of the Symmetric Part 13

To prove that Deﬁnition 1.2.1 (ii) is satisﬁed, we notice that according to Proposition 1.2.1 (ii), the function

t t t → E(Q u, Q u) is decreasing and convex, and hence

1 t t t+s t+s 1 t t E(Q u, Q u) − E(Q u, Q u) ≤ E(u, u) − E(Q u, Q u) s t for all s>0. Multiplying through by s,weget

t t t+s t+s s s s 0 ≤ E(Q u, Q u) − E(Q u, Q u) ≤ E(u, u) − E(Q u, Q u) . t For s ≈ 0 and t ≈ 0, the expression on the right is infinitesimal, and the lemma follows. ◦ We shall now strengthen the lemma above and show that if E1(u, u) < ∞, t then there is an infinitesimal t such that Q u ∈D(E). This is a special case of our next result. First we need to introduce a new definition. A subset F of H is called E-closed if and only if for all sequences {un}n∈N of elements from ◦ F such that |un − um|1 −→ 0 as n, m −→ ∞ , there exists an element u in F ◦ such that |un − u|1 −→ 0 as n −→ ∞ .

Proposition 1.2.3. D(E) is E-closed. Moreover, if {un}n∈N is a |·|1 Cauchy ∗ sequence from D(E),and{un | n ∈ N} is an internal extension, then there ∗ is a γ ∈ N − N such that uη ∈D(E) for all η ≤ γ.

Proof. Let {un|n ∈ N} be a |·|1 Cauchy sequence from D(E),andlet{un|n ∈ ∗N} be an internal extension of it. There is an element γ ∈ ∗N − N such that |un − um|1 ≈ 0 whenever n and m are inﬁnite and less than γ.Let ∗ ◦ ◦ η ∈ N − N,η ≤ γ. By the choice of γ, E1(uη,uη) < ∞ and |un − uη|1 −→ 0 as n approaches inﬁnity in N. All that remains is to prove that uη ∈D(E).

Assume not, then by Proposition 1.2.2 (iii) there is an ε ∈ R+ and t ≈ 0 such that

t |uη − Q uη|0 >ε.

Choose m ∈ N so large that

| − | ε uη um 0 < 4.

Then by Proposition 1.2.1 (i), we have

| t − t | ε Q uη Q um 0 < 4. 14 1 Hyperﬁnite Dirichlet Forms

Combining the inequalities above, we have

◦ t Corollary 1.2.3. If E1(u, u) < ∞,thereisat0 ≈ 0 such that Q u ∈D(E) for all t ≥ t0.

t0 t Proof. FirstwenoticethatifQ u ∈D(E),soisQ u for all t>t0.Put 1 n un = Q u. Then the sequence {|un|1} is increasing and bounded by |u|1, and we can apply Proposition 1.2.3 to it. The corollary follows. ◦ (δ) Corollary 1.2.4. If E1(u, u) < ∞,thereisaδu ≈ 0 such that u ∈D(E ) for all infinitesimal δ ≥ δu. t Proof. By Corollary 1.2.3,thereisat0 ≈ 0 such that Q u ∈D(E) for all t ≥ t0. Let δu be an infinitesimal such that δu >t0 and t0/δu ≈ 0. For all δ (δ) infinitesimal δ ≥ δu,wehavev = Q u ∈D(E) ⊂D(E ) by Corollaries 1.2.2 and 1.2.3. For all k ∈ ∗N such that kδ ≈ 0,wehavethefollowing

(δ) kδ kδ (δ) (k−1)δ (k−1)δ E (Q u, Q u)=E (Q v, Q v) (δ) ≈ E (v, v) (δ) δ δ = E (Q u, Q u). (1.2.7)

By Lemma 1.2.1 (iii), we have (δ) δ δ Δt s/2 δ s/2 δ E (Q u, Q u)= E(Q Q u, Q Q u) δ 0≤s<δ Δt s/2+δ−t0 t0 s/2+δ−t0 t0 = E(Q Q u, Q Q u) δ 0≤s<δ t t ≈ E(Q 0 u, Q 0 u), (1.2.8)

t because Q 0 u ∈D(E). By Lemma 1.2.1 (iii) again, we have (δ) Δt s/2 s/2 E (u, u)= E(Q u, Q u) δ 0≤s<δ ⎡ ⎤ Δt s/2 s/2 s/2 s/2 = ⎣ E(Q u, Q u)+ E(Q u, Q u)⎦ δ 0≤s<2t0 2t0≤s<δ 1.2 Domain of the Symmetric Part 15 Δt s/2−t0 t0 s/2−t0 t0 ≈ E(Q Q u, Q Q u) δ 2t0≤s<δ t t ≈ E(Q 0 u, Q 0 u). (1.2.9)

(δ) By relations (1.2.7), (1.2.8), and (1.2.9), we know u ∈D(E ). Remark 1.2.1. Proposition 1.2.3 is rather surprising since there exist standard forms which are neither closed nor closable. In fact, there are numerous applications where the main difficulty is to show that the form constructed is closed, or at least can be extended to a closed form (see, e.g., [11, 16– 24, 26, 27, 36, 48, 49, 94, 98, 99, 103, 151, 175, 176, 178, 225, 232, 236, 247, 251, 259, 278, 301, 318, 345, 359]). If we know that a form comes from a hyperfinite form, this follows immediately from Proposition 1.2.3. In Albeverio et al. [25], Chap. 6, we have got various examples of how useful this observation is. For the time being, we only remark that since we shall soon show that all standard, coercive closed forms can be obtained from hyperfinite forms, the method is quite general (we refer to Sect. 1.6 of this chapter). ◦ Notice that if we can show that whenever E1(u, u) < ∞, then for all t ≈ 0, t ||u − Q u|| ≈ 0, Corollary 1.2.3 will imply the first part of our program, i.e., (1.2.2)above. Lemma 1.2.4. Assume ◦E(u, u) < ∞. Then for all t ≈ 0, we have

t ||u − Q u|| ≈ 0.

Proof. For t ≈ 0, we have

t t t ||u − Q u||2 = u − Q u, u − Q u = E (t)( − t ) t u, u Q u (t) (t) t = t E (u, u) − E (u, Q u) ≤ tE(u, u) ≈ 0. Let us turn our attention to our second main goal (1.2.3). Lemma 1.2.5. If u, v ∈D(E) and u ≈ v,then

E(u, u) ≈E(v, v).

Proof. It is obviously enough to show that if u ∈D(E) and u ≈ 0,then E(u, u) ≈ 0. But if u ∈D(E), we know from Proposition 1.2.2 (iv):

(t) ◦E(u, u) = lim ◦E (u, u). (1.2.10) t↓0 t≈0 16 1 Hyperﬁnite Dirichlet Forms

Also

(t) 1 t E (u, u)= (I − Q )u, u t 1 t = u, u−Q u, u t 1 ≤ ||u||2, t

which is inﬁnitesimal for t ≈ 0. Combining this with (1.2.10), the lemma follows.

We may now sum up our results on D(E) in one statement.

Theorem 1.2.1. Let E(·, ·) be a nonnegative quadratic form on a hyperﬁnite dimensional space H.Then (i) If u, v ∈D(E) and α is a ﬁnite element of ∗R,thenαu, u + v ∈D(E). (ii) D(E) is E-closed. ◦ (iii) If E1(u, u) < ∞, then there exists a v ∈D(E) with ||u − v|| ≈ 0. Moreover, we have

t t ◦E(v, v) = lim ◦E(Q u, Q u) t↓0 t≈0 (t) = lim ◦E (u, u). t↓0 t≈0

(iv) If u, v ∈D(E) and u ≈ v, then E(u, u) ≈ E(v, v). (v) If u ∈D(E),then◦E(u, u)=inf{◦E(v, v) | v ≈ u}. (vi) If ◦E(u, u) < ∞ and ◦E(u, u)=inf{◦E(v, v) | v ≈ u}, then u ∈D(E).

Proof. We only need to show (v) and (vi), since we have proved the other results.

(v) Noticing (iv), we have E(u, u) ≈ 0 if u ∈D(E) and u ≈ 0. This implies the following for general u ∈D(E):

inf{◦E(v, v) | v ≈ u}≤◦E(u, u) 2 ◦ ◦ ≤ inf E(v, v)+ E(u − v, u − v) v ≈ u

=inf{◦E(v, v) | v ≈ u}.

This is (v).

t t (vi) From Proposition 1.2.1, we know that t → E(Q u, Q u) is decreasing. This implies (vi). 1.3 Resolvent of the Symmetric Part 17

The following deﬁnition now makes sense.

Deﬁnition 1.2.2. The symmetric standard part of E(·, ·) is the quadratic form E(·, ·) on ◦H deﬁned by: (i) The domain D(E) of E(·, ·) is the set of all equivalence classes ◦u ∈ ◦H ◦ ◦ such that inf{ E 1(v, v) | v ∈ u} < ∞. (ii) If x, y ∈ ◦H are in the domain of E(·, ·),letE(x, y)=◦E(u, v), where u ∈ x, v ∈ y are in D(E).

For α ∈ [0, ∞),letusset

Eα(·, ·)=E(·, ·)+α(·, ·).

We recall that (·, ·) is the inner product of ◦H.

An E1-Cauchy sequence is a sequence {xn} of elements from D(E) such that E1(un − um,un − um) −→ 0 as n, m −→ ∞ . We say that E(·, ·) is closed if all E1-Cauchy sequences converge in E1-norm to an element in D(E). The next proposition follows immediately from Theorem 1.2.1 and the deﬁnition of E(·, ·).

Proposition 1.2.4. Let E(·, ·) be the standard part of E(·, ·).ThenE(·, ·) is closed, and for all x ∈ ◦H

E(x, x)=inf{◦E(u, u) | u ∈ x}, (1.2.11) wherewetakethevalue∞ on the right to mean that the expression on the left is undeﬁned.

We point out that (1.2.11) is just our original suggestion (1.2.1)forthe standard part of E(·, ·). In Sect. 1.4, we shall study the standard part E(·, ·) of E(·, ·) under the hyperﬁnite weak sector condition. We shall show that D(E)=D(E) and that E(·, ·) is exactly the symmetric part of E(·, ·), i.e., 1 E(x, y)= 2 (E(x, y)+E(y,x)).

1.3 Resolvent of the Symmetric Part

In Sects. 1.1 and 1.2, we have only been interested in the relationship among the form E(·, ·) and the associated semigroup {Qt} and inﬁnitesimal generator t A,alsoE(·, ·) and its semigroup {Q } and generator A,andsoon.Inthis section, we turn our attention to the resolvent {Gα} of E(·, ·). The goal is to give a description of E(·, ·) and D(E) in terms of {Gα}, similar to the one we have given using the semigroup. The main result (Theorem 1.3.1) will allow us to reconstruct a form from its resolvent. One may want to notice that the 18 1 Hyperﬁnite Dirichlet Forms

result of Theorem 1.3.1 has played an essential part in the study of singular perturbations of operators in Albeverio et al. [25], Chap. 6.

−1 The operator Gα is deﬁned to be (A−α) whenever this exists. A formal calculation

−1 −1 A − α = I − I − Δt(A − α) Δt ∞ k = I − Δt(A − α) Δt k=0 ∞ Δt k = Q + αΔt Δt (1.3.1) k=0

tells us that Gα will exist if the series on the right hand side converges. Δt Since Q is a nonnegative and symmetric operator with norm at most one, all its eigenvalues must be between zero and one. We get that the absolute Δt value of all eigenvalues of Q + αΔt must be less than 1+αΔt. Hence if α<0 and |α|Δt < 2, the series in (1.3.1) converges, and we get the following proposition.

Proposition 1.3.1. Let E(·, ·) be a nonnegative quadratic form. Then −1 ∗ (i) (A − α) = Gα exists for all α ∈ (−∞, 0). Moreover, we have ||Gα|| ≤ 1 |α| in operator norm. ∗ 1 (ii) For α ∈ R, − Δt <α<0, we have

∞ Δt k Gα = Q + αΔt Δt. (1.3.2) k=0

Proof. (i) We notice that for all α ∈ ∗(−∞, 0), u ∈ H

(A − α)u, u≥−αu, u.

−1 ∗ This implies (A − α) = Gα exists for all α ∈ (−∞, 0) by elementary linear algebra. Furthermore, we have

αGαu, αGαu≤−αE−α(Gαu, Gαu)=−αu, Gαu≤||u||||αGαu||.

1 This implies that ||Gα|| ≤ |α| .

(ii) We have already proved this point before stating our proposition.

∗ We call {Gα | α ∈ (−∞, 0)} the resolvent of E(·, ·). 1.3 Resolvent of the Symmetric Part 19

In the standard Dirichlet space theory, the formula corresponding to (1.3.2)is

∞ ∞ −t(A−α) −tA αt Gα = e dt = e e dt, 0 0

−tA giving Gα as a weighted sum of the elements e in the semigroup. Since − ∞ αt =1 − 0 αe dt , it is convenient to multiply this equation by α to obtain

∞ −tA αt − αGα = − αe e dt. (1.3.3) 0

It is not quite obvious that this result carries over to the hyperﬁnite set- Δt kΔt ting, since the equation (Q + αΔt)k = Q (1 + αΔt)k (corresponding to e−t(A−α) = e−tA · eαt) is false. But the next result shows that the two operators are close enough for our purposes.

Lemma 1.3.1. ∈ ∗R − √1 ≤ 0 ∈ ◦E ( ) For α , Δt α< , and all u H with 1 u, u < ∞, we have ∞ kΔt k − 1+ ≈ 0 αGαu α Q αΔt Δt u . (1.3.4) k=0 1

Proof. Let {ei | 1 ≤ i ≤ N} be an orthonormal basis of eigenvectors for A,andletai be the i-th eigenvalue. Deﬁning bi = ai +1,wenoticethatif N u = i=1 uiei,then

N 2 E1(u, u)= biui . (1.3.5) i=1

Summing geometric series, we see that ∞ k αGα(ei)= α 1 − Δtai + Δtα Δt ei k=0 α = ei ai − α and similarly ∞ kΔt k α α Q 1+αΔt Δt (ei)= ei. i − + i k=0 a α a αΔt 20 1 Hyperﬁnite Dirichlet Forms

This yields ∞ kΔt k αGα(u) − α Q 1+αΔt Δt (u) k=0 N aiΔtuiei = . i=1 1 − (ai/α) 1 − (ai/α) − aiΔt

Taking the |·|1-norm of this, we get from (1.3.5) ∞ 2 kΔt k − 1+ αGαu α Q αΔt Δt u k=0 1 N 2 2 2 ai Δt = biui 2 2 i=1 1 − (ai/α) 1 − (ai/α) − aiΔt

≤ ηE1(u, u), where ⎛ ⎞ ⎜ a2Δt2 ⎟ η =max⎝ i ⎠ . 1≤i≤N 2 2 1 − (ai/α) 1 − (ai/α) − aiΔt

All that remains is to show that η is infinitesimal. First we observe that since ||A||Δt ≤ 1 (recalling (1.1.3) in Sect. 1.1), we have aiΔt ≤ 1 for all i. Hence ⎧ ⎫ ⎪ ⎪ ⎨ 2 2 ⎬ ai Δt η ≤ max 2 1≤i≤N ⎩⎪ 2 ⎭⎪ 1 − (ai/α) (ai/α) ⎧ ⎫ ⎨⎪ ⎬⎪ α4Δt2 =max . 1≤i≤N ⎩⎪ 2 ⎭⎪ α − ai √ Since −1/ Δt ≤ α<0, the latter term is infinitesimal, and the proof is finished. Equation (1.3.4) is the nonstandard counterpart of (1.3.3). Notice that

∞ k (−αΔt) 1+αΔt =1, k=0 and that if α is inﬁnite, then there is tα ≈ 0 such that 1.3 Resolvent of the Symmetric Part 21

k (−αΔt) 1+αΔt ≈ 1. (1.3.6)

0≤kΔt≤tα

On the other hand, if α is ﬁnite, then

k (−αΔt) 1+αΔt ≈ 1 t≤kΔt 0≤k<∞

for all inﬁnitesimal t. We can now begin our description of E(·, ·) and D(E) in terms of Gα.

Lemma 1.3.2. − √1 0 ◦E( ) ∞ If Δt <α< and u, u < , then

(i) If α is infinite, we have || − αGαu − u|| ≈ 0. (ii) If α is finite, we have −αGαu ∈D(E). (iii) There is an infinite α such that −αGαu ∈D(E).

Proof. According to Lemma 1.3.1, it suﬃces to prove the statements we get after replacing Gα by

∞ kΔt k Rα = Q 1+αΔt Δt. k=0

(i) Let tα ≈ 0 be as in (1.3.6). Then, we have ∞ k kΔt −αRαu − u = Q u − u (−α) 1+αΔt Δt k=0 k kΔt ≈ Q u − u (−α) 1+αΔt Δt 0

where the last step uses Lemma 1.2.4. t (ii) Choose t ≈ 0 such that Q u ∈D(E). If {ei}i≤N is an orthonormal basis of eigenvectors for A,andai is the i-th eigenvalue, we have

kΔt k k k Q 1+αΔt Δtei = 1 − aiΔt 1+αΔt Δtei 0≤kΔt

N If u = i=1 uiei,wegetfrom(1.3.5) 22 1 Hyperﬁnite Dirichlet Forms 2 kΔt (1 + )k Q u αΔt Δt 0≤kΔt

t/Δt t/Δt 1 − 1+αΔt 1 − aiΔt ≈ 0 ai − α + aiαΔt for all i. Hence, we have kΔt k 1+ ≈ 0 Q u αΔt Δt . (1.3.7) 0≤kΔt

kΔt We also remark that since Q u ∈D(E) for all kΔt ≥ t, we must have

kΔt k Q u 1+αΔt Δt ∈D(E). t≤kΔt

kΔt k But by the relation (1.3.7), we have |Rαu− t≤kΔt Q u 1+αΔt Δt|1 ≈

0. Hence, we have Rαu ∈D(E).

(iii) We remark that

N 2 E (− − )= 2 α 1 αGαu, αGαu biui ( − )2 i=1 ai α N 1 = 2 biui ( − 1)2 (1.3.8) i=1 ai/α increases as α −→ − ∞ , and is bounded by E1(u, u). Applying Proposition 1.2.3 to the sequence un = nG−nu, the lemma follows.

The next proposition adds two new characterizations of D(E) to the list in Proposition 1.2.2

Proposition 1.3.2. The following statements are equivalent: (i) u ∈D(E). ◦ ◦ (ii) E1(u, u) < ∞ and lim◦α−→ − ∞ E 1(u + αGαu, u + αGαu)=0. 1.3 Resolvent of the Symmetric Part 23

◦ ◦ (iii) E1(u, u) = lim◦α−→ − ∞ E 1(−αGαu, −αGαu) < ∞.

Proof. (i)=⇒ (ii). Pick an inﬁnite α such that −αGαu ∈D(E).Thenu + αGαu ∈D(E),u+ αGαu ≈ 0, and hence E1(u + αGαu, u + αGαu) ≈ 0. Part (ii) follows.

(ii) =⇒ (iii). By (1.3.8) and the triangle inequality

0 ≤|u|1 −|−αGαu|1 ≤|u + αGαu|1,

and multiplying by |u|1 + |−αGαu|1 ≤ 2E1(u, u), we get

2 2 0 ≤|u|1 −|−αGαu|1 ≤ 2E1(u, u) ·|u + αGαu|1,

which shows that (ii)=⇒ (iii).

(iii) =⇒ (i). Pick an inﬁnite α such that −αGαu ∈D(E).Then u + αGαu ≈ 0 and E1(u, u) ≈E1(−αGαu, −αGαu), and hence u ∈D(E). The following results gives a way of reconstructing a form from its resolvent. In the study of singular perturbations in Albeverio et al. [25], Chap. 6, we have found it much easier to control the resolvent of the perturbed form than the form itself. Once we have a good grasp of the resolvent, Theorem 1.3.1 will give us the form. Theorem 1.3.1. Let E(·, ·) be a nonnegative hyperﬁnite form on H,andlet E(·, ·) be its symmetric standard part. For all x ∈ ◦H and all v ∈ x, we have

◦ 2 E(x, x)=− lim α Gαv, v + αv, v . (1.3.9) ◦α−→ − ∞

Proof. Notice that since Gα is bounded, it does not matter which v ∈ x we use. We split the proof into two cases.

(i) x is not in the domain of E(·, ·): Let {ei | i ≤ N} be an orthonormal basis of eigenvectors for A, and assume that the corresponding eigenvalues {ai}i≤N are in decreasing order. Pick v = i≤N viei in x.Aneasy calculation shows that N − − 2 + = 2 α α Gαv, v α v, v aivi − . i=1 ai α

Assume for contradiction that the limit in (1.3.9) is finite. Then there is an infinite α such that ◦ N − 2 α ∞ aivi − < . i=1 ai α 24 1 Hyperfinite Dirichlet Forms

If H is the largest integer such that aH > |α|, we have

N H N 2 −α 2 −α 2 −α aivi = aivi + aivi i − i − i − i=1 a α i=1 a α i=H+1 a α H N 2 1 2 1 = −α vi + aivi 1 − i 1 − i i=1 α/a i=H+1 a /α H 1 N ≥−α 2 + 2 2 vi 2 aivi . i=1 i=H+1 α H 2 1 N 2 Hence, the last two terms − v and aiv are finite. But 2 i=1 i 2 i=H+1 i H 2 if −(α/2) i=1 vi is finite, v is infinitely close to

N v = viei. i=H+1 1 N 2 ◦ In addition, if 2 i=H+1 aivi is ﬁnite, then E1(v ,v ) < ∞. This contradicts the assumption that x ∈ D(E).

(ii) Assume that x ∈ D(E):Let v ∈ x be such that

◦ E 1(v, v) < ∞. (1.3.10)

Since E−α(Gαu, w)=u, w, we have

E(−αGαv, −αGαv)=E−α(αGαv, αGαv)+ααGαv, αGαv

2 3 = α Gαv, v + α Gαv, Gαv.

The theorem will follow from Proposition 1.3.2 (iii) if we can prove that for all v satisfying the condition (1.3.10),

◦ 2 3 2 lim α Gαv, v + α Gαv, Gαv + α Gαv, v + αv, v =0. ◦α→∞

By simple algebra, this is the same as

◦ 2 lim α||αGαv + v|| =0. ◦α→−∞

Pulling α inside the norm and reformulating the problem in nonstandard terms,weseethatwhatwehavetoproveis 1.3 Resolvent of the Symmetric Part 25 2 3/2 1 |α| Gαv −|α| 2 v ≈ 0 (1.3.11) for all inﬁnite, negative α of suﬃciently small absolute value. N If v = i=1 viei is the eigenvector expansion of v,weseethat 2 N | |3/2 2 | |3/2 −| |1/2 = α −| |1/2 2 α Gαv α v − α vi i=1 ai α N − 2 = αai 2 ( − )2 vi i=1 ai α N 2 1 = aiv , (1.3.12) i + −1 +2 i=1 βi βi where βi = −α/ai.

Notice that if ai is infinitesimal compared to α or α is infinitesimal com- −1 pared to ai,then1/(βi +βi +2) is infinitesimal. To get the sum on the right hand side of (1.3.12) to be infinitesimal, we only have to choose α such that the contributions from the terms satisfying neither of these requirements are infinitesimal.

Assuming that the eigenvalues {ai} are given in descending order, we define ◦ k 2 γ =sup aivi ak is infinite . i=1 ◦ k 2 ◦ Since ( i=1 aivi ) ≤ E(v, v) is finite by (1.3.10), γ is a real number. Using saturation8 on the sets j ∗ 2 1 An = j ∈ N aivi >γ− and aj >n , i=1 n we find a hyperinteger K such that aK is infinite and

K 2 aivi ≈ γ. i=1

8 Saturation is also a term of nonstandard analysis, we refer to Albeverio et al. [25]. 26 1 Hyperﬁnite Dirichlet Forms

We choose |α| to be inﬁnitely large, but inﬁnitesimal compared to aK .

For each ε ∈ R+,let k = 2 ≥ + Mε inf k aivi γ ε . i=1

By our choice of γ,thetermaMε must be ﬁnite. But

n K Mε−1 2 1 2 1 2 aiv ≤ aiv + aiv i + −1 +2 i + −1 +2 i i=1 βi βi i=1 βi βi i=K+1 N 2 1 + aivi −1 , βi + β +2 i=Mε i where the first term is infinitesimal since each βi is; the second term is less than 2ε by our choice of Mε; and the last term is infinitesimal since each βi is infinite. Since ε ∈ R+ is arbitrary, the sum on the left must be infinitesimal. This proves the approximation (1.3.11). Hence, the theorem is also proved.

1.4 Weak Coercive Quadratic Forms

In Sect. 1.3, we have discussed the resolvent of the symmetric part E(·, ·) of E(·, ·). We have heavily depended on the eigenvectors and eigenvalues of the generator A. In this section, we shall study the resolvent of E(·, ·) directly. However, the generator A is not symmetric. This forces us to ﬁnd an alternative way for the discussion.

Let E(·, ·) be a nonnegative quadratic form on a hyperﬁnite dimensional linear space H.LetA and Aˆ be the inﬁnitesimal generator and co-generator of E(·, ·), respectively. For α ∈ ∗( −∞, 0),u∈ H,wehave

(A − α)u, u≥−αu, u and (Aˆ − α)u, u≥−αu, u. (1.4.1)

−1 ˆ −1 ˆ ∗ Hence, (A − α) = Gα and (A − α) = Gα exist for all α ∈ ( −∞, 0) by elementary linear algebra. Moreover, we have

Proposition 1.4.1. Let E(·, ·) be a nonnegative quadratic form on a hyper- ﬁnite dimensional linear space H.Then −1 ˆ −1 ˆ ∗ (i) (A − α) = Gα and (A − α) = Gα exist for all α ∈ ( −∞, 0). 1 ˆ 1 Moreover, we have ||Gα|| ≤ |α| and ||Gα|| ≤ |α| . 1.4 Weak Coercive Quadratic Forms 27

(ii) For all α, β ∈ ∗( −∞, 0), we have

Gα − Gβ =(α − β)GαGβ =(α − β)GβGα (1.4.2)

and ˆ ˆ ˆ ˆ Gα − Gβ =(α − β)GαGβ ˆ ˆ =(α − β)Gβ Gα. (1.4.3)

−1 ˆ −1 ˆ Proof. (i) From (1.4.1), we know (A − α) = Gα and (A − α) = Gα exist for all α ∈ ∗( −∞, 0). Besides, we have

αGαu, αGαu≤−αE−α(Gαu, Gαu) = −αu, Gαu ≤||u||||αGαu||.

1 ˆ 1 This implies that ||Gα|| ≤ |α| . Similarly, we have ||Gα|| ≤ |α| .

(ii) We notice that

α − β =(A − β) − (A − α).

Hence, we have

(α − β)(A − α)−1 =(A − α)−1(A − β) − I.

This implies

(α − β)(A − α)−1(A − β)−1 =(A − α)−1 − (A − β)−1.

Therefore, we have

(α − β)GαGβ = Gα − Gβ.

Similarly, we can show that

(α − β)GβGα = Gα − Gβ.

This proves the relation (1.4.2). In the same manner, we can prove the relation (1.4.3).

ˆ Thereafter, we shall call {Gα | α<0} the resolvent of E(·, ·), and {Gα | α<0} the co-resolvent of E(·, ·).Therelation(1.4.2) will be called the ﬁrst 28 1 Hyperﬁnite Dirichlet Forms

resolvent equation,andtherelation(1.4.3) will be called the ﬁrst co-resolvent equation.

For α<0, we deﬁne

(α) E(u, v)=−αu + αGαu, v ˆ = −αv + αGαv, u,u,v∈ H,

and (α) ˆ E(u, v)=−αv + αGαv, u ˆ = −αu + αGαu, v,u,v ∈ H.

Then

(α) E(u, −αGαu)=−αu + αGαu, −αGαu (α) = E(u, u)+αu + αGαu, u + αGαu,u∈ H,

and (α) ˆ ˆ ˆ E(−αGαu, u)=−αu + αGαu, −αGαu (α) ˆ ˆ = E(u, u)+αu + αGαu, u + αGαu,u∈ H.

Therefore, we get

(α) (α) E(u, −αGαu) ≤ E(u, u),u∈ H, (1.4.4)

and (α) ˆ (α) E(−αGαu, u) ≤ E(u, u),u∈ H.

Actually, we have

Lemma 1.4.1. Let E(·, ·) be a nonnegative quadratic form on a hyperﬁnite dimensional linear space H.Then (α) (α) ˆ ˆ (i) E(u, v)=E(−αGαu, v) and E(u, v)=E(−αGαu, v) for all u, v ∈ H. (α) ˆ ˆ (α) (ii) E(−αGαu, −αGαu) ≤ E(u, u) and E(−αGαu, −αGαu) ≤ E(u, u) for all u ∈ H.

Proof. (i) We have

(α) 2 E(u, v)=−αu, v−α Gαu, v 2 = −αE−α(Gαu, v) − α Gαu, v = E(−αGαu, v). 1.4 Weak Coercive Quadratic Forms 29

(ii) It follows from (i) and (1.4.4)that

(α) E(−αGαu, −αGαu)= E(u, −αGαu) ≤ (α)E(u, u).

Notice that ˆ ˆ E(−αGαu, −αGαu)=E(−αGαu, −αGαu).

This implies (ii).

Definition 1.4.1. Let E(·, ·) be a nonnegative quadratic form on a hyperfinite dimensional linear space H.WecallE(·, ·) a hyperfinite weak coercive ∗ ◦ quadratic form if and only if there exists a constant C ∈ R+ with 0 ≤ C<∞ such that |E1(u, v)|≤C E1(u, u) E1(v, v), ∀u, v ∈ H. (1.4.5)

We call (1.4.5)thehyperfinite weak sector condition and C a continuity constant. Lemma 1.4.2. Let E(·, ·) be a nonnegative quadratic form on a hyperfinite dimensional linear space H. Then the following statements are equivalent: (i) E(·, ·) satisfies the hyperfinite weak sector condition. ∗ ◦ ∗ ◦ (ii) For β ∈ (0, ∞), ∞ > β>0, there exists Cβ ∈ R+ with 0 ≤ Cβ < ∞ such that

|Eβ(u, v)|≤Cβ Eβ(u, u) Eβ(v, v), ∀u, v ∈ H.

∗ ◦ ∗ ◦ (iii) For β ∈ (0, ∞), ∞ > β>0, there exists Cβ ∈ R+ with 0 ≤ C β < ∞ such that

|E(u, v)|≤Cβ Eβ(u, u) Eβ(v, v), ∀u, v ∈ H.

Proof. The proof is easy and is left as an exercise. Lemma 1.4.3. Let E(·, ·) be a hyperfinite weak coercive quadratic form on a hyperfinite dimensional linear space H. Then for all α ∈ ∗( −∞, 0) (α) (α) (i) | E1(u, v)|≤(C1 +1) E1(u, u) E1(v, v) for all u, v ∈ H. 2 (ii) E1(−αGαu, −αGαu) ≤ (C1 +1) E1(u, u) for all u ∈ H. Proof. (i) By using Lemma 1.4.1 and Lemma 1.4.2,wehave (α) ˆ E(u, v) = E(u, −αGαv) ˆ ˆ ≤ C1 E1(u, u) E1(−αGαv, −αGαv) (α) ≤ C1 E1(u, u) E(v, v). 30 1 Hyperfinite Dirichlet Forms

(ii) From (i) and Lemma 1.4.1 (ii), we have

(α) E1(−αGαu, −αGαu) ≤ E(u, u) 2 ≤ (C1 +1) E1(u, u).

Lemma 1.4.4. Let E(·, ·) be a hyperfinite weak coercive quadratic form on a ◦ hyperfinite dimensional linear space H.If E1(u, u) < ∞, then for all infinite α<0, we have ||u + αGαu|| ≈ 0.

Proof. From Lemma 1.4.1 (i) and Lemma 1.4.3 (i), we have

2 ||u + αGαu|| = u + αGαu, u + αGαu (α)E( + ) = − u, u αGαu α 1 (α) (α) = − E(u, u)+ E(u, αGαu) α 1 (α) = − E(u, u) −E(αGαu, αGαu) α (α)E( ) ≤− u, u α 2 (C1 +1) ≤− E1(u, u) α ≈ 0.

It is the time to introduce the deﬁnition of the domain D(E) of E(·, ·).

Definition 1.4.2. Let E(·, ·) be a nonnegative quadratic form on a hyperfinite dimensional linear space H. The domain D(E) of E(·, ·) is the set of all u ∈ H satisfying ◦ (i) E 1(u, u) < ∞. ˆ (ii) For all infinite α<0, E(u + αGαu, u + αGαu) ≈ 0 and E(u + αGαu, u + ˆ αGαu) ≈ 0.

Proposition 1.4.2. Let E(·, ·) be a hyperfinite weak coercive quadratic form on a hyperfinite dimensional linear space H. Then the following statements are equivalent: (i) u ∈D(E). ◦ (ii) E1(u, u) < ∞, and for all infinite α<0, E(u + αGαu, u) ≈ 0 and ˆ E(u + αGαu, u) ≈ 0. ◦ (α) (iii) E1(u, u) < ∞, and for all infinite α<0, E(u, u) ≈E(u, u) and (α)Eˆ(u, u) ≈E(u, u). 1.4 Weak Coercive Quadratic Forms 31

Proof. (i)=⇒ (ii). This is easily seen from Lemma 1.4.2 (iii) and Lemma 1.4.4.

(ii) =⇒ (i). From Lemma 1.4.1,wehave

0 ≤E(u + αGαu, u + αGαu) = E(u, u + αGαu)+E(αGαu, u + αGαu) (α) = E(u, u + αGαu)+E(−αGαu, −αGαu) − E(u, u) ≤E(u, u + αGαu) ˆ = E(u + αGαu, u) ≈ 0.

Similarly, we can show ˆ ˆ E(u + αGαu, u + αGαu) ≈ 0.

(ii) ⇐⇒ (iii). This is easily seen from Lemma 1.4.1. Lemma 1.4.5. Let E(·, ·) be a hyperfinite weak coercive quadratic form on a hyperfinite dimensional linear space H.If◦E(u, u) < ∞, then for all finite ◦ β<0, β =0, Gβu ∈D(E). Proof. We have from Lemma 1.4.3 (ii) that ◦ 2 (C1 +1) E1(Gβ u, Gβu) ≤ E1(u, u) < ∞. β2

Hence, it suﬃces to show that for all inﬁnite α<0 by Proposition 1.4.2 (ii)

E(Gβu + αGαGβu, Gβu) ≈ 0 (1.4.6) and ˆ E(Gβ u + αGαGβu, Gβu) ≈ 0. (1.4.7)

Actually, we have

E(Gβ u + αGαGβu, Gβu) = u + αGαu, Gβu + βGβu + αGαGβu, Gβu. (1.4.8)

From Lemma 1.4.4,weseethat

u + αGαu, Gβu≈0. (1.4.9)

From the relation (1.4.2), we have 32 1 Hyperﬁnite Dirichlet Forms α βGβ u + αGαGβu, Gβu = βGβ u + (Gα − Gβ)u, Gβu α − β −α α = β(1 + )Gβu + Gαu, Gβu α − β α − β −β2 αβ = Gβu, Gβu + Gαu, Gβu α − β α − β ≈ 0. (1.4.10)

By the relations (1.4.8), (1.4.9), and (1.4.10), we have proved the relation (1.4.6). Similarly, we can prove the relation (1.4.7). We recall that the norm |·|1 is deﬁned by |u|1 = E1(u, u). A subset F of H is called E-closed if and only if for all sequences {un}n∈N of elements ◦ from F such that |un − um|1 −→ 0 as n, m −→ ∞ , there exists an element ◦ u in F such that |un − u|1 −→ 0 as n −→ ∞ . One may want to notice that a subset F of H is E-closed if and only if it is E-closed.

As we just mention above, the reader may already note that the definitions of E-closedness and E-closedness are formally identical. However, in the definition of E-closedness, the norm |·|1 is defined via |u|1 = E(u, u),whereas in the definition of E-closedness, it is defined via |u|1 = E(u, u). It will always be clear from the context which norm we actually mean by |·|1. Proposition 1.4.3. Let E(·, ·) be a hyperfinite weak coercive quadratic form on a hyperfinite dimensional linear space H.ThenD(E) is E-closed. More- ∗ over, if {un}n∈N is a |·|1 Cauchy sequence from D(E),and{un | n ∈ N} ∗ is an internal extension of {un}n∈N, then there is a γ ∈ N − N such that uη ∈D(E) for all η ≤ γ.

Proof. Let {un | n ∈ N} be a |·|1 Cauchy sequence from D(E),andlet ∗ ∗ {un | n ∈ N} be an internal extension of it. There is an element γ ∈ N − N such that |un − um|1 ≈ 0 whenever n and m are inﬁnite and less than γ.Let ∗ ◦ ◦ η ∈ N − N,η ≤ γ. By the choice of γ, E1(uη,uη) < ∞ and |un − uη|1 −→ 0 as n approaches inﬁnity in N. All that remains is to prove that uη ∈D(E).

Assuming not, there is an ε ∈ R+ and inﬁnite β<0 such that

|uη + βGβuη|1 >ε (1.4.11) or ˆ |uη − βGβ uη|1 >ε. (1.4.12)

(1) Assume that the relation (1.4.11) holds. Choose m ∈ N so large that

| − | ε uη um 1 < 2 . 4(C1 +1) 1.4 Weak Coercive Quadratic Forms 33

Then by Lemma 1.4.3 (ii), we have

| − | ε βGβ uη βGβ um 1 < 4.

Combining the inequalities above, we get

ε<|uη + βGβuη|1 ≤|uη − um|1 + |um + βGβum|1 + |βGβ um − βGβuη|1 ≤ ε + | + | 2 um βGβum 1.

However, the last term |um + βGβum|1 is inﬁnitesimal by Lemma 1.4.4 since um ∈D(E). We have the contradiction we wanted.

(2) If the relation (1.4.12) holds, then we can get a corresponding contradiction. As in Sect. 1.2, we may now sum up our results on D(E) in one statement. Theorem 1.4.1. Let E(·, ·) be a hyperfinite weak coercive quadratic form on a hyperfinite dimensional space H.Then (i) If u, v ∈D(E) and α is a finite element of ∗R,thenαu, u + v ∈D(E). (ii) D(E) is E-closed. ◦ (iii) If E1(u, u) < ∞, then there exists a v ∈D(E) with ||u − v|| ≈ 0. Moreover, we have

◦ ◦ E(v, v) = lim E(αGαu, αGαu) ◦α↓−∞ ◦α=−∞ = lim ◦[(α)E(u, u)]. ◦α↓−∞ ◦α=−∞

(iv) If u, v ∈D(E) and u ≈ v, then E(u, u) ≈E(v, v). (v) If u ∈D(E),then◦E(u, u)=inf{◦E(v, v) | v ≈ u}. (vi) If ◦E(u, u)=inf{◦E(v, v) | v ≈ u} < ∞,thenu ∈D(E). Proof. (i) We can prove this in the same way as the proof of Corollary 1.2.1. (ii) It is proved in Proposition 1.4.3.

◦ (iii) Since E 1(u, u) < ∞, we have from Lemma 1.4.3 (ii) that for all α ∈ ∗( −∞, 0) ◦ ◦ 2 [E1(−αGαu, −αGαu)] ≤ (C1 +1) E1(u, u) < ∞.

◦ Let {−αnGαn } be a |·|1 Cauchy sequence, α ↓−∞.Let{−αnGαn | n ∈ ∗N} be an internal extension of it. There exists an inﬁnite η ∈ ∗N such 34 1 Hyperﬁnite Dirichlet Forms

that −αnGαn u ∈D(E),n≤ η. By Lemma 1.4.4,wehaveu ≈−αηGαη u.

Hence by letting v = −αηGαη u, we have gotten (iii).

(iv) It is obviously enough to show that if u ∈D(E) and u ≈ 0,then E(u, u) ≈ 0. But if u ∈D(E), we know from (iii):

◦E(u, u) = lim (α)[◦E(u, u)]. (1.4.13) ◦α↓−∞ ◦α=−∞

Also

(α) E(u, u)=−αu + αGαu, u 2 = −αu, u−α Gαu, u ≤−α||u||2,

which is inﬁnitesimal for α ≈−∞. Combining this with (1.4.13), (iv) follows.

(v) This follows in the same way as for the proof of Theorem 1.2.1 (v).

(vi) For simplicity, we assume that u ≈ 0.ThenE(u, u) ≈ 0. From Lemma 1.4.3 (i), we have for all α ∈ ∗(−∞, 0)

(α) 0 ≤| E1(u, u)| 2 ≤ (C1 +1) E1(u, u) ≈ 0.

(α) ∗ Hence, E1(u, u) ≈ 0, ∀α ∈ (−∞, 0). Similarly, we can show that (α) ˆ E 1(u, u) ≈ 0. By Proposition 1.4.2,wehaveu ∈D(E).

The following deﬁnition now makes sense.

Definition 1.4.3. Let E(·, ·) be a hyperfinite weak coercive quadratic form on a hyperfinite dimensional space H.Thestandard part of E(·, ·) is the quadratic form E(·, ·) on ◦H defined by: (i) The domain D(E) of E(·, ·) is the set of all equivalence classes ◦u ∈ ◦H ◦ ◦ such that inf{ E1(v, v) | v ∈ u} < ∞. (ii) If x, y ∈ ◦H are in the domain of E(·, ·),letE(x, y)=◦E(u, v), where u ∈ x, v ∈ y are in D(E).

For α ∈ [0, ∞),letusset

Eα(·, ·)=E(·, ·)+α(·, ·).

We recall that (·, ·) is the inner product of ◦H. 1.4 Weak Coercive Quadratic Forms 35

Proposition 1.4.4. Let E(·, ·) be a hyperﬁnite weak coercive quadratic form on a hyperﬁnite dimensional space H.LetE(·, ·) be the standard part of E(·, ·). Then E(·, ·) is closed, and for all x ∈ ◦H

E(x, x)=inf{◦E(u, u) | u ∈ x}, (1.4.14) wherewetakethevalue∞ on the right to mean that the expression on the left is undeﬁned.

◦ Proof. If E 1(u, u)=∞ for all u ∈ x,itiseasytosee(1.4.14) holds. Assume that inf{◦E(u, u) | u ∈ x} < ∞, then E(x, x)=◦E(v, v) for some v ∈ D(E),v ∈ x. Hence, we only need to show ◦E(v, v)=inf{◦E(u, u) | u ∈ x}. But this is implied by Theorem 1.4.1 (iv).

In Sect. 1.2, we have gotten the symmetric standard part E(·, ·) of E(·, ·). We have studied the domain D(E) of E(·, ·) also. Now it is very natural to discuss the relation between the results of Sect. 1.2 and those of this section. Actually, we have

Theorem 1.4.2. Let E(·, ·) be a hyperﬁnite weak coercive quadratic form on a hyperﬁnite dimensional space H.ThenD(E)=D(E) and D(E)=D(E). Moreover, we have 1 ( )= ( )+ ( ) E x, y 2 E x, y E y,x . (1.4.15)

Proof. From Theorem 1.4.1 (v) and (vi), we know that v ∈D(E) if and only if ◦E(v, v)=inf{◦E(u, u) | v ≈ u} and ◦E(v, v) < ∞. This statement is also true for the elements in D(E) from Theorem 1.2.1 (v) and (vi). Hence, D(E)=D(E).Therelation(1.4.15) follows immediately.

Remark 1.4.1. Let E(·, ·) be a hyperfinite weak coercive quadratic form on a hyperfinite dimensional space H. It is easy to see that (E(·, ·),D(E)) satisfies the following weak sector condition |E1(x, y)|≤C E1(x, x) E1(y,y) for all x, y ∈ D(E), where C ∈ R+ is a positive real number. Hence, (E(·, ·),D(E)) is a coercive closed form on ◦H (the definition of coercive closed form will be given in Sect. 1.6). 36 1 Hyperfinite Dirichlet Forms 1.5 Hyperfinite Dirichlet Forms

It is time to discuss the hyperﬁnite forms associated with Markov processes, i.e., the Dirichlet forms. The aim is to give a reasonably detailed account of the relationship between the properties of these forms and the behavior of the associated processes.

Consider a particle which can be in N +1 different states s0,s1,s2, ··· ,sN . Assume that if the particle is in state si at some instant t, then – independently of what its past history may be – the probability that it will be in state sj at the next instant t + Δt is given by a fixed number qij . This is the familiar setting for the theory of stationary Markov chains with finite state space, see, e.g., Chung [119], Dynkin [147], and Dynkin and Yushkevich [148]. We shall be interested in the case where S = {s0,s1, ··· ,sN } is a hyperfinite ∗ set, and T = {kΔt | k ∈ N0} is a hyperfinite time line with Δt ≈ 0 (for technical reasons it is convenient to have an ∗-infinite time line to work with). The idea is to use the hyperfinite setup to reduce the highly sophisticated theory of continuous parameter Markov processes taking values in topological spaces (see [96,175,193,270,330,333,334]) to the much simpler theory of finite Markov chains.

Let Y be a Hausdorﬀ space, i.e., Y is a topological space such that for each pair x, y of distinct points in Y ,thereareopensetsU, V satisfying x ∈ U, y ∈ V, and U ∩ V = ∅ [121]. Here ∅ denotes the empty set. Let ∗Y be the nonstandard extension of Y .LetS = {s0,s1, ··· ,sN } be an S-dense subset of ∗Y for some N ∈ ∗N − N and m be a hyperﬁnite measure on S. Denote by S the internal algebra of subsets of S. Assume that Q = {qij } is an (N +1)× (N +1)matrix with nonnegative entries, and assume that

N qij =1 for all i =0, 1, ··· ,N, (1.5.1) j=0 and the state s0 is a trap, i.e.,

q0i =0 for all i =0. (1.5.2)

In the sequel, we shall write mi for m({si}) and qij for qsisj respectively, whenever it is convenient.

If (Ω,P) is an internal measure space, and X : Ω × T −→ S is an internal process, let

[ω]t = {ω ∈ Ω | X(ω ,s)=X(ω,s) for all s ≤ t}. (1.5.3)

For each t ∈ T, let Ft be the internal algebra on Ω generated by the sets [ω]t. 1.5 Hyperﬁnite Dirichlet Forms 37

If for all ω ∈ Ω

P ([ω]0)=m{X(ω,0)}, (1.5.4) and whenever X(ω,t)=si,

P {ω ∈ [ω]t | X(t + Δt, ω )=sj} = qij P ([ω]t), (1.5.5) then we call X a hyperfinite Markov chain with initial distribution m and transition matrix Q. Notice that we do not assume that m and P are probability measures. P (Ω) could be an infinite, hyperfinite number.

In fact, given m and Q, it is easy to construct an associated Markov chain X(ω,t).LetΩ be the set of all internal functions ω : T −→ S. Denote by X the coordinate function X(ω,t)=ω(t).LetP be the measure deﬁned by k−1 P [ω]kΔt = m({ω(0)}) qω(nΔt),ω((n+1)Δt). (1.5.6) n=0

In particular, we define a family (Ω,Ft,Pi,i ∈ S) of internal probability spaces by k−1 Pi [ω]kΔt = δiω(0) qω(nΔt),ω((n+1)Δt) (1.5.7) n=0 for each i ∈ S, where δij is the Kronecker symbol. ˆ Similarly, let Q = {qîj } be an (N +1)× (N +1)matrix with nonnegative entries, and assume that N qîj =1 for all i =0, 1, ··· ,N, (1.5.8) j=0 and the state s0 is a trap, i.e.,

qˆ0i =0 for all i =0. (1.5.9)

In the same manner as above, we shall write qˆij for qˆsisj , respectively, whenever it is convenient.

If (Ω,ˆ Pˆ) is an internal measure space, and Xˆ : Ωˆ × T −→ S is an internal process, let

ˆ ˆ ˆ [ˆω]t = {ωˆ ∈ Ω | X(ˆω ,s)=X(ˆω,s) for all s ≤ t}. ˆ ˆ For each t ∈ T, let Ft be the internal algebra on Ω generated by the sets [ˆω]t. 38 1 Hyperﬁnite Dirichlet Forms

If for all ωˆ ∈ Ωˆ ˆ ˆ P ([ˆω]0)=m{X(ˆω,0)}, (1.5.10) ˆ and whenever X(ˆω,t)=si,

ˆ ˆ ˆ P{ωˆ ∈ [ˆω]t | X(t + Δt, ωˆ )=sj} =ˆqij P ([ˆω]t), (1.5.11) then we call Xˆ a hyperﬁnite Markov chain with initial distribution m and transition matrix Qˆ.

Given m and Qˆ, we can construct an associated Markov chain Xˆ(ˆω,t) as ˆ ˆ ˆ that of (1.5.6). Moreover, we deﬁne a family (Ω,Ft, Pi,i ∈ S) of internal probability spaces by

k−1 ˆ Pi [ˆω]kΔt = δiωˆ(0) qˆωˆ(nΔt),ωˆ((n+1)Δt) (1.5.12) n=0 for each i ∈ S. It is easy to see that we can take ˆ ˆ ˆ Ω = Ω, X = X, [ω]t =[ˆω]t, Ft = Ft. ˆ ˆ ˆ However, Q, P ,andPi are diﬀerent from Q, P ,andPi.

Now let us introduce some regularity conditions. We assume that the measure m and the transition matrices Q and Qˆ satisfy the dual conditions

miqij = mjqˆji for all i =0,j =0. (1.5.13)

Besides, we assume that

mi =0 for at least one i =0. (1.5.14)

It is easy to ﬁnd examples of transition matrices Q and Qˆ such that no m satisﬁes the conditions (1.5.13)and(1.5.14). Thus, these assumptions may be regarded as conditions on Q and Qˆ.

Notice that for most i, the transition probabilities qi0 and qˆi0 should be of order of magnitude Δt, since if not the process will die in inﬁnitesimal time.

Given m, Q and Qˆ which satisfy conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14), the processes X and Xˆ as above are called dual hyperfinite Markov chains. These are the processes 1.5 Hyperfinite Dirichlet Forms 39 we will study in detail. We shall first associate to a given process a hyperfinite quadratic form.

S0 = {s1,s2, ··· ,sN } is the state space S without the trap so,letussetS0 = S∩S0.LetH be the ∗ linear space of all internal functions u : S0 −→ R with the inner product uv dm = u, v S0 N = u(si)v(si)m(si). (1.5.15) i=1

Just as we usually write mi for m(si), we shall write u(i) or ui for u(si).And we shall identify H with the set of all internal functions u : S −→ ∗R such that u(s0)=0.

Our convention of letting the trap s0 be the zeroth element is notationally convenient, but we call attention of the reader to the fact that she/he should N N distinguish between sums of the forms i=0 and i=1,e.g.

For t ∈ T and u ∈ H, we deﬁne new functions Qtu, Qˆtu ∈ H by

t Q u(i)=Eiu(X(t)), ˆt ˆ Q u(i)=Eiu(X(t)), (1.5.16) ˆ where Ei and Ei are the expectations with respect to the measures Pi and ˆ t ˆt Pi deﬁned in (1.5.7)and(1.5.12), respectively. Intuitively, Q u(i) and Q u(i) are the expected values of u(X(t)) for a particle starting in state si.Notice that

QΔtu(i)=(Q · u)(i) N = u(j)qij , j=1

QˆΔtu(i)=(Qˆ · u)(i) N = u(j)ˆqij , j=1 40 1 Hyperﬁnite Dirichlet Forms where · are the matrix multiplications in the middle terms. Since

N (t+s) (t) (s) qij = qik qkj , k=1 N (t+s) (t) (s) qˆij = qˆik qˆkj , k=1 we must have

Qt+s = Qt · Qs, Qˆt+s = Qˆt · Qˆs,

(t) (t) t where qij and qˆij are the transition probabilities given by operator Q and Qˆt, respectively. Hence, the families {Qt | t ∈ T } and {Qˆt | t ∈ T } are semigroups of operators on H. Actually, {Qˆt | t ∈ T } is the co-semigroup of {Qt | t ∈ T }.

The inﬁnitesimal generator A of the semigroup {Qt | t ∈ T } is given by

1 N Au(i)= u(i) − u(j)qij . (1.5.17) Δt j=1

The hyperﬁnite quadratic form associated with Q and m is deﬁned to be

E(u, v)=Au, v N = Au(i)v(i)m(i). (1.5.18) i=1

Combining (1.5.17)and(1.5.18), we get ⎡ ⎤ N N 1 ⎣ ⎦ E(u, v)= u(i)v(i)m(i) − u(j)v(i)qij m(i) . (1.5.19) Δt i=1 j=1

The inﬁnitesimal co-generator Aˆ of the co-semigroup {Qˆt | t ∈ T } is given by

N ˆ 1 Au(i)= u(i) − u(j)ˆqij . (1.5.20) Δt j=1 1.5 Hyperﬁnite Dirichlet Forms 41

The hyperﬁnite quadratic co-form associated with Qˆ and m is deﬁned to be

Eˆ(u, v)=E(v, u) = Av, u N = Av(i)u(i)m(i). (1.5.21) i=1

Combining (1.5.13), (1.5.17), (1.5.20), and (1.5.21), we get ⎡ ⎤ N N ˆ 1 ⎣ ⎦ E(u, v)= u(i)v(i)m(i) − u(i)v(j)qij m(i) Δt i=1 j=1 ⎡ ⎤ N N 1 ⎣ ⎦ = u(i)v(i)m(i) − u(i)v(j)ˆqjim(j) Δt i=1 j=1 = Au,ˆ v. (1.5.22)

t Now let us look at the symmetric part E(·, ·) of E(·, ·).Let{Q | t ∈ T } be the semigroup and A be the generator of E(·, ·). Then, we have 1 = ( +ˆ ) qij 2 qij qij ,

Δt where (qij )=Q = Q . It is easy to see that

miqij = mjqji.

Moreover, the generator A of E(·, ·) is given by 1 ( )= ( )+ ˆ ( ) Au i 2 Au i Au i 1 N 1 = ( ) − ( ) ( +ˆ ) u i u j 2 qij qij Δt j=1 1 N = u(i) − u(j)qij . Δt j=1

The symmetric hyperﬁnite quadratic form E(·, ·) is given by 42 1 Hyperﬁnite Dirichlet Forms N E(u, v)= Au(i)v(i)m(i) i=1 ⎡ ⎤ N N 1 ⎣ ⎦ = u(i)v(i)m(i) − u(j)v(i)qij m(i) . (1.5.23) Δt i=1 j=1

Our first result in the following lemma gives alternative ways of expressing hyperfinite quadratic forms in term of mi, qij , qîj and qij . They are nonstandard versions of the Beurling–Deny formulae [87, 88], and are often more useful than the expressions (1.5.19), (1.5.22), and (1.5.23).

Lemma 1.5.1. Let E(·, ·) be the hyperﬁnite quadratic form as above. Then, we have

1 N (i) E(u, v)= u(i) − u(j) v(i)qij mi + u(i)v(i)qi0mi . Δt 1≤i,j≤N i=1 (1.5.24) 1 N (ii) E(u, v)= v(i) − v(j) u(i)ˆqij mi + u(i)v(i)ˆqi0mi . Δt 1≤i,j≤N i=1 (1.5.25) 1 (iii) E(u, v)= u(i) − u(j) v(i) − v(j) qij m(i) Δt 1≤i

Proof. We only prove (iii) since the other proofs are similar. Notice that ⎡ ⎤ N N N 1 ⎣ ⎦ E(u, v)= u(i)v(i)mi − u(j)v(i)qij mi Δt i=1 i=1 j=1 ⎡ ⎤ N N N N 1 ⎣ ⎦ = u(i)v(i)qij mi − u(j)v(i)qij mi Δt i=1 j=0 i=1 j=1 ⎡ ⎤ N 1 ⎣ ⎦ = u(i) − u(j) v(i)qij mi + u(i)v(i)qi0mi , Δt 1≤i,j≤N i=1 where the first line is a trivial modification of the expression (1.5.23), the N second line follows from the first since j=0 qij =1, and the last line is just a rearrangement of the second line. 1.5 Hyperfinite Dirichlet Forms 43

Fix a pair (i, j) and consider the terms in the last expression above involving both i and j.Ifi = j, there is only one such term, and that term is zero. If i = j, there are two terms to consider, i.e.,

(u(i) − u(j))v(i)qij mi and (u(j) − u(i))v(j)qjimj.

Since qij mi = qjimj, the sum of the two terms equals

(u(i) − u(j))(v(i) − v(j))qij mi.

Summing over all pairs (i, j), the relation (1.5.26) follows.

As an immediate consequence, we have: Corollary 1.5.1. The hyperfinite quadratic form E(·, ·) associated with Q and m as above is nonnegative. In order to get some further properties of the hyperfinite quadratic form E(·, ·), let us introduce the following definitions.

For u ∈ H, v ∈ H, we deﬁne

(u ∨ v)(s)=max{u(s),v(s)} and (u ∧ v)(s)=min{u(s),v(s)},s∈ S0.

Moreover, let u+ = u ∨ 0 and u− = −u ∧ 0.

If u ∈ H, the function u˜ =(0∨ u) ∧ 1 is called the unit contraction of u. A quadratic form E(·, ·) is said to have the Markov property if for all u

E(˜u, u˜) ≤E(u, u).

Corollary 1.5.2. Let E(·, ·) be the hyperﬁnite quadratic form as above. Then (1) For all u ∈ H, α ∈ ∗R,α≥ 0, we have

E(u ∧ α, u − u ∧ α) ≥ 0, E(u − u ∧ α, u ∧ α) ≥ 0.

(2) E(˜u, u˜) ≤E(u, u). Proof. (1) By the expression (1.5.24), we have 1 E(u ∧ α, u − u ∧ α)= u(i) ∧ α − u(j) ∧ α u − u ∧ α (i)qij mi Δt 1≤i,j≤N N + (u ∧ α)(i) u(i) − u(i) ∧ α qi0mi . (1.5.27) i=1 44 1 Hyperﬁnite Dirichlet Forms

If u(i) ≥ α, then

u(i) ∧ α − u(j) ∧ α u − u ∧ α (i)= α − u(j) ∧ α u(i) − α ≥ 0 (1.5.28) and

u ∧ α (i) u(i) − u(i) ∧ α = α u(i) − α ≥ 0. (1.5.29)

If u(i) <α,then

u(i) ∧ α − u(j) ∧ α u − u ∧ α (i)=(u ∧ α)(i) u − u ∧ α (i) =0. (1.5.30)

It follows from the relations (1.5.27), (1.5.28), (1.5.29), and (1.5.30)that E(u ∧ α, u − u ∧ α) ≥ 0. Similarly, we can show that E(u − u ∧ α, u ∧ α) ≥ 0 by using the relation (1.5.25).

(2) From the relation (1.5.26), we have E(˜u, u˜) ≤ E(u, u). This implies the conclusion (2), since E(u, u)=E(u, u) for all u ∈ H.

We call a map T : H −→ H a Markov operator, if it maps nonnegative functions into nonnegative functions, and never increases the supremum norm, i.e.,

||Tu||∞ ≤||u||∞ for all u ∈ H, where ||u||∞ =max1≤i≤N |u(i)|.

The next result gives us four ways of deciding whether a given quadratic form is a hyperﬁnite quadratic form associated with some Q and m without actually constructing an associated Markov process.

Proposition 1.5.1. Let

N E(u, v)= bij u(i)v(j) i,j=1 be a nonnegative quadratic form which is non-zero. The following statements are equivalent: 1.5 Hyperﬁnite Dirichlet Forms 45

(i) E(·, ·) is the hyperﬁnite quadratic form of some Q, Qˆ,andm which satisfy the conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14). (ii) There exist a hyperﬁnite measure m on S0 and two Markov operators QΔt, QˆΔt : H −→ H such that

QΔtu, v = u, QˆΔtv

and 1 E(u, v)= (I − QΔt)u, v Δt 1 = u, (I − QˆΔt)v, Δt

where (H, ·, ·)isdeﬁnedthrough(1.5.15)bym,andΔt is an inﬁnites- imal. (iii) For all u ∈ H, α ∈ ∗R,α≥ 0, we have

E(u ∧ α, u − u ∧ α) ≥ 0, E(u − u ∧ α, u ∧ α) ≥ 0.

(iv) E(·, ·) satisﬁes E(˜u, u − u˜) ≥ 0 and E(u − u,˜ u˜) ≥ 0, ∀u ∈ H,whereu˜ is the unit contraction of u. (v) Whenever i = j, bij ≤ 0; but bii ≥− j =i bij and bii ≥− j =i bji for all i. Proof. We shall prove (i)=⇒ (iii)=⇒ (iv)=⇒ (v)=⇒ (i)=⇒ (ii)=⇒ (v).

(i) =⇒ (iii). This follows immediately from Corollary 1.5.2.

(iii) =⇒ (iv). We have

E(˜u, u − u˜)=E(˜u, (u ∨ 0) − u˜)+E(˜u, u − (u ∨ 0)) = E((u ∨ 0) ∧ 1, (u ∨ 0) − (u ∨ 0) ∧ 1) + Eu(˜u, − u ∨ 0) ≥E(˜u, u − u ∨ 0) = E((u ∧ 1)+, −(u ∧ 1)−).

Let v = u ∧ 1,then

E(˜u, u − u˜) ≥E((u ∧ 1)+, −(u ∧ 1)−) = −E(v+,v−) = −E(v+,v+ − v) = E((−v) ∧ 0, (−v) − (−v) ∧ 0) ≥ 0. 46 1 Hyperﬁnite Dirichlet Forms

Similarly, we can show

E(u − u,˜ u˜) ≥ 0.

(iv) =⇒ (v). Fix k, l ∈ S0,k = l, deﬁne u by ⎧ ⎪ ⎨⎪−1 if i = k, ( )= 1 = u i ⎪ if i l, ⎩⎪ 0 otherwise.

If u˜ is the unit contraction of u,wehave

0 ≤E(˜u, u) −E(˜u, u˜) = bll − blk − bll = −blk.

Therefore, blk ≤ 0.

To get the second half of (v), we ﬁx a k ∈ S0 and consider the function 2 if i = k, v(i)= 1 otherwise

If v˜ is the unit contraction of v,then

0 ≤E(˜v,v − v˜) N = bik i=1

and

0 ≤E(v − v,˜ v˜) N = bkj. j=1

(v) =⇒ (i). We ﬁrst observe (v(i) − v(j))u(i)bij = u(i)v(i)bij − u(i)v(j)bij 1≤i,j≤N 1≤i,j≤N 1≤i,j≤N = u(i)v(i)bij −E(u, v). 1≤i,j≤N 1.5 Hyperﬁnite Dirichlet Forms 47

Thus, we have E(u, v)=− (v(i) − v(j))u(i)bij + u(i)v(i)bij . (1.5.31) 1≤i,j≤N 1≤i,j≤N

Similarly, we have (u(i) − u(j))v(i)bji = u(i)v(i)bji − u(j)v(i)bji 1≤i,j≤N 1≤i,j≤N 1≤i,j≤N = u(i)v(i)bji −E(u, v). 1≤i,j≤N

Hence, we have E(u, v)=− (u(i) − u(j))v(i)bji + u(i)v(i)bji. (1.5.32) 1≤i,j≤N 1≤i,j≤N

In the following, we shall deﬁne Q and m by marching the two expressions (1.5.25)and(1.5.31), (1.5.24)and(1.5.32) term by term. It turns out that we have one degree of freedom for each i.Wemaychoose

0 ≤ qii < 1,qii =ˆqii.

Comparing (1.5.25)and(1.5.31), (1.5.24)and(1.5.32), we notice that we must have 1 miqˆij = −bij Δt 1 = qjimj for j =0,i (1.5.33) Δt and

1 N miqˆi0 = bij , Δt j=1 1 N miqi0 = bji. (1.5.34) Δt j=1

N In order to have j=0 qîj =1satisfied, we must have 48 1 Hyperfinite Dirichlet Forms N mi = miqîj j=0 N = miqîi − Δt bij + Δt bij j =i,0 j=1

= miqˆii + biiΔt.

Hence, this gives us

biiΔt mi = 1 − qii biiΔt = . 1 − qˆii

Combining this with (1.5.33), we get

bji(1 − qii) qij = − , bii bij (1 − qii) qˆij = − for j =0,i, bii and from (1.5.34), we have

N j=1 bij (1 − qˆii) qˆi0 = , bii N j=1 bji(1 − qii) qi0 = . bii

This construction breaks down when bii =0, but in that case bij =0for all j,andwegetmi =0and can choose the qij ’s arbitrarily.

We ﬁnally observe that (1.5.33) implies (1.5.13), and that the non-triviality of E(·, ·) implies (1.5.14).

(i) =⇒ (ii). This follows immediately from (1.5.16).

(ii) =⇒ (v). Notice that

N 1 Δt ( )= ( ) − ( ) Q u i u i Δt bjiu j ( ), j=1 m i

Δt for m(i) =0.Form(i)=0, we simplify here Q u(i)=u(i).Foreachj ∈ S0, Δt let uj be given by uj(i)=δij .SinceQ is Markov, we have for all i = j 1.5 Hyperﬁnite Dirichlet Forms 49

Δt 0 ≤ Q uj(i) 1 = −Δtbji , m(i) and thus bji ≤ 0.

On the other hand, applying QΔt to the function which is constant one and using that QΔt cannot increase the supremum norm, we get

1 ≥ QΔt1(i) N 1 =1− ( ) =0 Δt bji ( ), if m i , j=1 m i Δt and Q 1(i)=1if m(i)=0. It follows that bii ≥− j =i bji. Besides, we have

N 1 ˆΔt ( )= ( ) − ( ) Q u i u i Δt bij u j ( ). j=1 m i

Hence, we have

1 ≥ QˆΔt1(i) N 1 =1− Δt bij ( ). j=1 m i From this, we have bii ≥− j =i bij .

Now it is the time to introduce the following: Definition 1.5.1. The hyperfinite quadratic form E(·, ·) associated with Q and m is called a hyperfinite Dirichlet form if it satisfies the hyperfinite weak sector condition in the sense of Definition 1.4.1.

If δ ∈ T ,letTδ be the sub-line

∗ Tδ = {kδ | k ∈ N0} .

(δ) (δ) We write X for the restriction X|Tδ. For each t ∈ Tδ,letFt be the internal algebra on Ω generated by the sets (δ) (δ) (δ) [ω]t = ω ∈ ΩX (ω ,s)=X (ω,s) for all s ∈ Tδ,s≤ t . (1.5.35) 50 1 Hyperﬁnite Dirichlet Forms

It is easy to verify the following for all k ∈ ∗N,

k−1 (δ) (δ) P [ω]kδ = m(ω(0)) qω(nδ),ω((n+1)δ) n=0 and k−1 (δ) (δ) Pi [ω]kδ = δiω(0) qω(nδ),ω((n+1)δ). n=0

Therefore, for any t ∈ Tδ and u ∈ H,wehave

(δ) t Eiu(X (t)) = Q u(i) = Eiu(X(t)).

In particular, we get

(δ) δ Eiu(X (δ)) = Q u(i) N (δ) = qij u(j). j=1

(δ) t This implies that the semigroup of X is {Q | t ∈ Tδ}. The inﬁnitesimal generator A(δ) of X(δ) is given by

N (δ) 1 (δ) A u(i)= u(i) − u(j)qij . δ j=1

ˆ (δ) ˆ(δ) (δ) ˆt ˆ(δ) Similarly, we can get X , Ft , [ˆω]t , {Q | t ∈ Tδ}, A . It is easy to see that A(δ) and Aˆ(δ) are nonnegative, i.e.,

A(δ)u, u≥0 and Aˆ(δ)u, u≥0 for all u ∈ H.

Moreover, we can easily show that

N N (δ) (δ) qij =1and qˆij =1for all i =0, 1, 2 ··· ,N, j=0 j=0

(δ) (δ) q0i =0and qˆ0i =0 for all i =0 1.5 Hyperﬁnite Dirichlet Forms 51 and

(δ) (δ) miqij = mjqˆji for all i =0,j =0.

The hyperﬁnite quadratic form associated with Q(δ) and m is deﬁned to be

E(δ)(u, v)=A(δ)u, v N = A(δ)u(i)v(i)m(i). i=1

Therefore, we have ⎡ ⎤ 1 N N (δ) ⎣ (δ) ⎦ E (u, v)= u(i)v(i)m(i) − u(j)v(i)qij m(i) δ i=1 j=1 ⎡ ⎤ N 1 ⎣ (δ) (δ) ⎦ = u(i) − u(j) v(i)qij m(i)+ u(i)v(i)qi0 m(i) . δ 1≤i,j≤N i=1

(δ) (δ) −1 ˆ(δ) ˆ(δ) −1 As in Sect. 1.4,letGα =(A − α) and Gα =(A − α) be the resolvents of A(δ) and Aˆ(δ),α∈ ∗(−∞, 0), respectively. Similarly, the domain D(E(δ)) of E(δ)(·, ·) is the set of all u ∈ H such that ◦ ◦ (δ) (δ) (i) E1 (u, u)= E (u, u)+u, u < ∞. (α) (δ) (δ) (ii) For all inﬁnitesimal α<0,wehave E(u+αGα u, u+αGα u) ≈ 0 and (α) ˆ(δ) ˆ(δ) E(u + αGα u, u + αGα u) ≈ 0.

ˆ(δ) (δ) (δ) t Moreover, we can deﬁne E (·, ·) and E (·, ·), A , {Q | t ∈ Tδ} and (α) D(E ) as in Sects. 1.1 and 1.2. We can get similar results of the Beurling– Deny formulae for these hyperﬁnite quadratic forms; in addition, we may get similar results as Proposition 1.5.1 for E(δ)(·, ·).

Let us try to illustrate the theory by a simple example.

Example 1.5.1. (Brownianmotiononacircle).LetN =(Δt)−1/2 be an even hyperfinite integer, and let S0 = {s1, ··· ,sN } be uniformly distributed on a circle of circumference one. If i, j ∈{1, 2, ··· ,N}, let the transition 1 probability qij be 2 if si and sj are neighbors, and 0 otherwise. The semigroup {Qt} is given by 1 1 Δt ( )= ( +1)+ ( − 1) Q u i 2u i 2u i , 52 1 Hyperfinite Dirichlet Forms where the addition is modulo N inside the u’s. The infinitesimal generator is

u(i +1)− 2u(i)+u(i − 1) Au(i)=− . (1.5.36) 2Δt

1 If mi = N for all i, the associated hyperﬁnite quadratic form E(·, ·) is given by N E( )=−N ( +1)− 2 ( )+ ( − 1) ( ) u, v 2 u i u i u i v i , i=1 or – in the Beurling–Deny formulation –

N E( )=N ( +1)− ( ) ( +1)− ( ) u, v 2 u i u i v i v i . (1.5.37) i=0

Notice that by (1.5.36), the inﬁnitesimal generator A is a nonstandard version of the operator 1 ˜ = − Af 2f , and by (1.5.37), the form E(·, ·) is a representation of 1 E(f,g)= f gdm, 2 C where m is the Lebesgue measure on the circle C, and all derivatives are taken along the circle. Passing from the original expression for E(·, ·) to the Beurling–Deny version amounts to an integration by parts.

1.6 Hyperﬁnite Representations

In this section, we will relate the results we have obtained so far to the standard Dirichlet space theory. We have proved that a hyperfinite weak coercive quadratic form on a hyperfinite dimensional space H induces a closed form on the hull ◦H of H. For most applications, the space ◦H is too large. What we really want is a form defined on a Hilbert space K given in advance of the nonstandard construction. If K can be identified with a subspace of ◦H, we get the desired form by restricting the form on ◦H to K. Noticing that since K is a closed subspace of ◦H, the restricted form is also closed. The result we shall prove in this section states that any coercive closed form on any Hilbert space K can be obtained from a nonnegative quadratic form in this way. There are two reasons for proving such a representation theorem. The first and most important is to obtain the “correct” relationship between the 1.6 Hyperfinite Representations 53 already developed standard theory on the one hand, and the new hyperfinite theory on the other. In fact, we shall utilize this relation to solve the problem in classical theory, referring to Theorem 4.1.1 in Chap. 4. The second, closely related reason is to show that no generality is lost by working within the nonstandard framework.

Let K be a standard Hilbert space with an inner product (·, ·).Ahyperﬁ- nite dimensional subspace H of ∗K is called S-dense in ∗K if for all x ∈ K, there is a y ∈ H such that ||x − y|| ≈ 0. We recall that ≈ is the equivalent relation on H given by u ≈ v if and only if ||u − v|| ≈ 0, and that ◦u denotes the equivalence class of u under ≈.IfH is S-dense in ∗K,wecanidentifyK with a subspace of ◦H by identifying x and ◦u whenever ||x − u|| ≈ 0.

If E(·, ·) is a hyperfinite weak coercive quadratic form on H,weletE(·, ·) denote the standard part of E(·, ·) as defined in Definition 1.4.3,andwelet EK (·, ·) be the restriction of E(·, ·) to K. As mentioned above, our goal is to show that all coercive closed forms can be obtained in this way. Before we prove this, we shall recall a few results from the standard theory of quadratic forms ([270] is a convenient reference).

Let D(F ) be a linear subspace of K,andletF : D(F ) × D(F ) −→ R be a bilinear map which is nonnegative, i.e., F (x, x) ≥ 0 for all x ∈ D(F ).For α ∈ R+,weset

Fα(·, ·)=F (·, ·)+α(·, ·).

(F (·, ·),D(F )) is said to satisfy the weak sector condition if |F1(x, y)|≤C F1(x, x) F1(y,y) for all x, y ∈ D(F ), where C is positive real number which is called a continuity constant.

(F (·, ·),D(F )) is said to be closed if D(F ) is complete with respect to the norm F1(·, ·).

We say that (F (·, ·),D(F )) is a coercive closed form on K if and only if (1) D(F ) is a dense subspace of K. (2) (F (·, ·),D(F )) satisﬁes the weak sector condition. (3) (F (·, ·),D(F )) is a closed form. Proposition 1.6.1. Let (F (·, ·),D(F )) be a coercive closed form on K with continuity constant C. Then there exist unique strongly continuous contrac- ˆ tion resolvent {Rα | α ∈ (−∞, 0)} and co-resolvent {Rα | α ∈ (−∞, 0)} on K such that ˆ ˆ Rα(K), Rα(K) ⊂ D(F ) and F−α(Rαf,u)=(f,u)=F−α(u, Rαf) (1.6.1) 54 1 Hyperﬁnite Dirichlet Forms for all f ∈ K, u ∈ D(F ),α∈ (−∞, 0). In particular, we have ˆ (Rαf,g)=(f,Rαg) for all f,g ∈ K, ˆ i.e., Rα is the adjoint of Rα for all α ∈ (−∞, 0). Proof. We refer to Ma and Röckner [270], Chap. I, Theorem 2.8 and use negative value of α for the resolvents. We assume that (F (·, ·),D(F )) is a coercive closed form on K with continuity constant C.Let(L, D(L)) and (L,ˆ D(Lˆ)) be the generators of ˆ {Rα | α ∈ (−∞, 0)} and {Rα | α ∈ (−∞, 0)}, respectively. Then we have from Ma and Röckner [270], Chap. I, Corollary 2.10,

Rα(K)=D(L) ⊂ D(F ) and F (u, v)=(Lu, v), ∀u ∈ D(L),v ∈ D(F ) and ˆ ˆ ˆ ˆ Rα(K)=D(L) ⊂ D(F ) and F (v, u)=(Lu, v), ∀u ∈ D(L),v ∈ D(F ).

Deﬁne for α ∈ (−∞, 0),u,v∈ K,

(α) F (u, v)=−α(u + αRαu, v) ˆ = −α(v + αRαv, u) and

(α) ˆ F (u, v)=−α(v + αRαv, u) ˆ = −α(u + αRαu, v).

Then for all α ∈ (−∞, 0),u∈ K,wehave

(α) (α) F (u, −αRαu)= F (u, u)+α(u + αRαu, u + αRαu) and

(α) ˆ (α) ˆ ˆ F (−αRαu, u)= F (u, u)+α(u + αRαu, u + αRαu).

Therefore, we have for all α ∈ (−∞, 0),u∈ K,

(α) (α) F (u, −αRαu) ≤ F (u, u), (α) ˆ (α) F (−αRαu, u) ≤ F (u, u).

Proposition 1.6.2. Let (F (·, ·),D(F )) be a coercive closed form on K with continuity constant C>0. Then, we have 1.6 Hyperﬁnite Representations 55

(α) (α) ˆ ˆ (i) F (u, v)=F (−αRαu, v) and F (u, v)=F (−αRαu, v) for all u ∈ K, v ∈ D(F ). (α) ˆ ˆ (α) (ii) F (αRαu, αRαu) ≤ F(u, u) and F (αRαu, αRαu) ≤ F (u, u) for all u ∈ K and α ∈ (−∞, 0). (α) (α) (iii) | F 1(u, v)|≤(C +1) F1(u, u) F 1(v, v) for all u ∈ D(F ),v ∈ K. 2 (iv) F1(αRαu, αRαu) ≤ (C +1) F1(u, u) for all u ∈ D(F ).

Proof. We refer to Ma and Röckner [270], Chap. I, Lemma 2.11.

Proposition 1.6.3. Let (F (·, ·),D(F )) be a coercive closed form on K with continuity constant C>0.Then ∈ ∈ ( ) sup (α) ( ) ∞ (i) Let u K. Then u D F if and only if α∈(−∞,0) F u, u < . (ii) D(L)=Rα(K) is dense in D(F ) with respect to F1(·, ·). Moreover, we have for all u ∈ D(F )

lim F1(αRαu + u, αRαu + u)=0. α−→ − ∞

In particular, (F (·, ·),D(F )) is determined by {Rα | α ∈ (−∞, 0)} via (1.6.1) uniquely. (α) (iii) limα−→ − ∞ F (u, v)=F (u, v) for all u, v ∈ D(F ).

Proof. We refer to Ma and Röckner [270], Chap. I, Theorem 2.13.

Let (F (·, ·),D(F )) be a coercive closed form on K with continuity constant ˆ C>0.Denoteby{Tt | t ∈ [0, ∞)} and {Tt | t ∈ [0, ∞)} the strongly continuous contraction semigroups corresponding to {Rα | α ∈ (−∞, 0)} and ˆ {Rα | α ∈ (−∞, 0)}, respectively. Then, we have from Ma and Röckner [270], Chap. I, Theorem 2.8 that ˆ (Ttf,g)=(f,Ttg) for all f,g ∈ K, t > 0. ˆ Thereafter, {Rα | α ∈ (−∞, 0)} and {Rα | α ∈ (−∞, 0)} shall be called resolvent and co-resolvent of (F (·, ·),D(F )), respectively. Similarly, {Tt | t ∈ ˆ [0, ∞)} and {Tt | t ∈ [0, ∞)} shall be called semigroup and co-semigroup of (F (·, ·),D(F )), respectively. For t ∈ (0, ∞), we deﬁne

(t) 1 F (u, v)= (u − Ttu, v),u,v ∈ K. t As in Proposition 1.6.3,wehave

Proposition 1.6.4. Let (F (·, ·),D(F )) be a coercive closed form on K with continuity constant C>0.Then 56 1 Hyperﬁnite Dirichlet Forms

(t) (i) Let u ∈ K.Thenu ∈ D(F ) if and only if supt>0 F (u, u) < ∞. (t) (ii) For all u, v ∈ D(F ), we have limt↓0 F (u, v)=F (u, v). (iii) limt↓0 F1(u − Ttu, u − Ttu)=0for all u ∈ D(F ).

Proof. We refer to Albeverio et al. [9], Theorem 3.4.

Proposition 1.6.5. Let (F (·, ·),D(F )) be a coercive closed form on K with continuity constant C>0. Then, we have for n ∈ N and ∀f ∈ K

−n t Ttf = lim I + L f n−→ ∞ n n n = lim R−n/t f n−→ ∞ t and

−n ˆ t ˆ Ttf = lim I + L f n−→ ∞ n n n ˆ = lim R−n/t f. n−→ ∞ t

Proof. We refer to Pazy [299], Theorem 8.3, Chap. I, page 33.

∗ Let stK be the standard part map from K to K.

Theorem 1.6.1. Let (F (·, ·),D(F )) be a coercive closed form on a Hilbert space K, and let H be an S-dense, hyperﬁnite dimensional subspace of ∗K. Then, there exists a nonnegative quadratic form E(·, ·) on H – associated with ∗ an internal time line T = {0,Δt,2Δt, ··· ,kΔt,···}= {kΔt | k ∈ N0} with an inﬁnitesimal Δt > 0 –suchthat

F (·, ·)=EK (·, ·). (1.6.2)

∗ ˆ ∗ Moreover, if {Gα | α ∈ ( −∞, 0)} and {Gα | α ∈ ( −∞, 0)} are the resolvent and co-resolvent generated by E(·, ·), then for all β ∈ (−∞, 0),α ∈ ∗ ◦ ( −∞, 0),u∈ K, v ∈ H such that β = α, u = stK (v), we have ˆ ˆ stK Gαv = Rβu and stK Gαv = Rβu. (1.6.3)

On the other hand, if {Qs | s ∈ T } and {Qˆs | s ∈ T } are the semigroup and co-semigroup generated by E(·, ·), then for all t ∈ [0, ∞),s∈ T,u ∈ K, v ∈ H ◦ such that t = s, u = stK (v), we have

s ˆs ˆ stK Q v = Ttu and stK Q v = Ttu. (1.6.4) 1.6 Hyperﬁnite Representations 57

∗ ∗ Proof. Let P be the projection of K on H. Let us write { Rα}α∈∗(−∞,0) for ∗ ∗ ˆ ∗ ˆ ({Rα}α∈(−∞,0)),and{ Rα}α∈∗(−∞,0) for ({Rα}α∈(−∞,0)). Our plan is first Δt ∗ to define an internal semigroup by putting Q = −P (γRγ ) for a carefully 1 chosen infinitesimal Δt = − γ ,andthenlet

1 E(u, v)= (I − QΔt)u, v Δt ∗ = −γ(I + P (γRγ))u, v, (1.6.5) by using Proposition 1.6.3.Noticethatifu ∈ H,then

Δt ∗ Q u, u = −P (γRγ)u, u ∗ = − (γRγ)u, u ≥ 0.

Δt ∗ This shows that Q is positive on H. Also, since the operator norm of (γRγ ) is less than or equal to one, so is the norm of QΔt. Hence, the conditions in Sect. 1.1 are satisﬁed.

We shall now choose Δt such that the relations (1.6.2), (1.6.3), and (1.6.4) hold. If u ∈ H is nearstandard and −◦α<∞, then

∗ ∗ ∗ ˆ ∗ ˆ ||P (αRα)u − (αRα)u|| ≈ 0 and ||P (αRα)u − (αRα)u|| ≈ 0

∗ ∗ ˆ because (αRα) and (αRα) take the nearstandard elements to nearstandard elements, and because H is S-dense in ∗K. By induction, we get

∗ n ∗ n ∗ ˆ n ∗ ˆ n ||[P (αRα)] u − (αRα) u|| ≈ 0 and ||[P (αRα)] u − (αRα) u|| ≈ 0 for all n ∈ N.Foreachu ∈ K, let vu = Pu. Then vu ∈ H and ||u − vu|| ≈ 0. We consider the set ∗ 2n ∗ n k ∗ n k 1 Au = n ∈ N∀k ≤ 2 ||[P (2 R−2n )] vu − [ (2 R−2n )] u|| ≤ and n ∗ n ˆ k ∗ n ˆ k 1 ||[P (2 R−2n )] vu − [ (2 R−2n )] u|| ≤ . n (1.6.6)

For each u ∈ K, this set contains N, and hence an internal segment {n ∈ ∗ N | n ≤ nu}. By saturation, there is an inﬁnite n smaller than all the nu’s, u ∈ K. 58 1 Hyperﬁnite Dirichlet Forms

Next, we consider = ∈ ∗N∀ ≤ 22m | ( + ) Bu m k k I kR−k vu,vu ∗ 1 −k(I + P (kR ))vu,vu| ≤ . −k m

For each u ∈ K, this set contains N, and hence an internal segment {m ∈ ∗ N | m ≤ mu}. By saturation, there is an inﬁnite m smaller than all the mu’s, u ∈ K.

We now take Δt to be the inﬁnitesimal 2−m∨2−n.Thatis,Δt =2−m∨2−n, m n −1 −1 or γ = −2 ∧ 2 . Since Rα =(L − α) , we have L = α + Rα for all α ∈ (−∞, 0). Hence, we have for all u ∈ D(L)=R1(K),

−1 − α(I + αRα)u = −αRα(Rα + α)u = −αRα(Lu) −→ Lu, α −→ − ∞ ,α∈ (−∞, 0).

Therefore, we get for all inﬁnite α ∈ ∗( −∞, 0)

0 ≈||Lu + αRα(Lu)|| = ||Lu + α(I + αRα)u||, ∀u ∈ D(L). (1.6.7)

From the relations (1.6.5)and(1.6.7), we have for all u ∈ D(L)=R1(K) that

◦E(Pu,Pu)=F (u, u).

Because of the closedness of F (·, ·), we have got the following equation for all u ∈ D(F )

◦ E(Pu,Pu)=EK (u, u) = F (u, u).

This is (1.6.2). From (1.6.6) and Proposition 1.6.5,wehavetheresults(1.6.4).

In the following, we shall show that E(·, ·) satisﬁes (1.6.3). For all β ∈ ∗ ◦ (−∞, 0),α∈ ( −∞, 0),u∈ K, v ∈ H such that β = α, u = stK (v), we have

E−α(Gαv, Gαv − Rαvu)=v, Gαv − Rαvu 1.6 Hyperﬁnite Representations 59 and

E−α(Rαvu,Gαv − Rαvu) ∗ = −γ(I + P (γRγ ))Rαvu,Gαv − Rαvu−αRαvu,Gαv − Rαvu ≈−γ(I + γRγ))Rαvu,Gαv − Rαvu−αRαvu,Gαv − Rαvu ≈LRαvu,Gαv − Rαvu−αRαvu,Gαv − Rαvu ≈v, Gαv − Rαvu.

Hence, we have

|α|·||Gαv − Rαvu|| ≤ E−α(Gαv − Rαvu,Gαv − Rαvu) ≈ 0.

This implies the relations (1.6.3).

Now let us make a few comments on Theorem 1.6.1.

Remark 1.6.1. In the proof of Theorem 1.6.1, we apply Proposition 1.6.3 by Δt using the resolvent {Rα | α ∈ (−∞, 0)} to construct Q .Analternative way is to apply Proposition 1.6.4 by using the semigroup {Tt | t ∈ (0, ∞)} to construct QΔt, and the readers may refer to the proof of 5.2.1. Proposition, Albeverio et al. [25] for the details.

Remark 1.6.2. The assumption that (F (·, ·),D(F )) is densely defined is for convenience only. If it is not satisfied, we just apply the proposition to the closure of D(F ).IfF (·, ·) is not closed, we obviously cannot obtain F (·, ·) as EK (·, ·) for any hyperfinite form E(·, ·) since we need the closedness of F (·, ·) in the proof of Theorem 1.6.1. However, if F (·, ·) is closable (i.e., there exists a closed form extending F (·, ·)), all closed extensions of F (·, ·) can be repre- sented as standard parts of hyperfinite forms. A natural representation for a closed form F (·, ·) would be a representation of its smallest closed extension – the Friedrichs extension. If F (·, ·) is not closable, no hyperfinite representation (in our sense) is possible. Any representation we try will change some F (·, ·) values, and restrict and extend D(F ) in different directions in order to turn F (·, ·) into a closed form. As we commented in Sect. 1.2,thefact that non-closable forms do not have hyperfinite representation is more than a curse. In the standard theory, a lot of effort goes into showing that the forms one constructs are closable. In the hyperfinite theory, this is an immediate consequence of the construction.

Remark 1.6.3. In Theorem 1.6.1,thespaceH was just any S-dense, hyperfinite dimensional subspace of ∗K. In applications, we often want to choose special kind of subspaces which are appropriate for the problems we have in mind. In Chap. 4, we shall study the case K = L2(Y,ν) for some Hausdorff space Y . 60 1 Hyperfinite Dirichlet Forms

Remark 1.6.4. The lifting E(·, ·) for a coercive closed form F (·, ·) in Theo- rem 1.6.1 needs not satisfy the hyperfinite weak sector condition. Hence, E(·, ·) is not necessarily a hyperfinite weak coercive quadratic form. This makes it impossible for us to use the results in Sect. 1.4. However, we do have some good property about E(·, ·) which will be very useful in Chap. 4.Actually,we have the following from Proposition 1.6.2: Corollary 1.6.1. Let (F (·, ·),D(F )) be a coercive closed form on K with continuity constant C>0.Forα ∈ ∗(−∞, 0), we define for u, v ∈ ∗K

(α) F (u, v)=−α(u + αRαu, v) ˆ = −α(v + αRαv, u) and

(α) ˆ F (u, v)=−α(v + αRαv, u) ˆ = −α(u + αRαu, v).

Then (α) (α) ˆ ˆ (i) F (u, v)=F (−αRαu, v) and F (u, v)=F (−αRαu, v) for all u ∈ ∗K,v ∈ ∗(D(F )) and α ∈ ∗(−∞, 0). (α) ˆ ˆ (α) (ii) F (αRαu, αRαu) ≤ F(u, u) and F (αRαu, αRαu) ≤ F (u, u) for all ∗ ∗ u ∈ K and α ∈ (−∞, 0). (α) (α) ∗ (iii) | F 1(u, v)|≤(C +1) F1(u, u) F 1(v, v) for all u ∈ (D(F )),v∈ ∗K and α ∈ ∗(−∞, 0). 2 ∗ (iv) F1(αRαu, αRαu) ≤ (C +1)F1(u, u) for all u ∈ (D(F )) and α ∈ ∗(−∞, 0). Proof. The proof follows from Proposition 1.6.2. When are the standard forms generated by two hyperfinite forms different? The last result in this section we shall prove shows that to answer this question, it is enough to check whether the forms have the same resolvents. We recall that in Theorem 1.4.1 we found a way to construct a form from its resolvent. This representation will be helpful to solve our problem. Theorem 1.6.2. Let K be a Hilbert space and H be an S-dense, hyperfinite dimensional subspace of ∗K.LetE(·, ·) and E˘(·, ·) be two hyperfinite weak ˘ coercive quadratic forms on H inducing EK (·, ·) and EK (·, ·) on K, respec- ˘ ˘ tively. Let {Gα} and {Gα} be the resolvents of E(·, ·) and E(·, ·). Assume that for some finite, non-infinitesimal α ∈ ∗(−∞, 0), there is a u ∈ H with ◦ ˘ E1(u, u) < ∞ such that v = Gαu, w = Gαu are both nearstandard, but ◦ ˘ ||v − w|| =0. Then EK (·, ·) = EK (·, ·). ˘ Proof. Assume for contradiction that EK (·, ·)=EK (·, ·).Pickv˜ ≈ v, w˜ ≈ w such that v˜ ∈D(E˘), w˜ ∈D(E). Notice that by Lemma 1.4.5, v ∈D(E),w ∈ D(E˘). We have 1.7 Weak Coercive Quadratic Forms, Revisited 61

u, v − w≈u, v − w˜ = E−α(v, v − w˜). (1.6.8) ˘ Since v, w are nearstandard and EK (·, ·)=EK (·, ·), we have

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ E−α(v, v − w˜)=EK ( v, v − w) − ( α)( v, v − w) ˘ ◦ ◦ ◦ ◦ ◦ ◦ ◦ = EK ( v, v − w) − ( α)( v, v − w) ◦ ˘ = E−α(˜v, v˜ − w). (1.6.9)

On the other hand, we have

u, v − w≈u, v˜ − w ˘ = E−α(w, v˜ − w). (1.6.10)

Combining the relations (1.6.8), (1.6.9), and (1.6.10), we see that

◦ ˘ 0= E−α(˜v − w, v˜ − w) ≥ ◦|α|◦||v − w||2 > 0.

The theorem is proved.

1.7 Weak Coercive Quadratic Forms, Revisited

Let E(·, ·) be a hyperfinite weak coercive quadratic form on a hyperfinite ∗ dimensional space H.Let{Gα | α ∈ (−∞, 0)} be the resolvent of E(·, ·),and ˆ ∗ let {Gα | α ∈ (−∞, 0)} be the co-resolvent of E(·, ·), respectively. Let us still denote by A the infinitesimal generator of E(·, ·), and by Aˆ the infinitesimal co-generator of E(·, ·).SuchasinSect.1.1, we fix an infinitesimal Δt,andwe define new operators QΔt and QˆΔt by

QΔt = I − ΔtA, QˆΔt = I − ΔtA.ˆ

∗ Introduce a nonstandard time line T by T = {kΔt | k ∈ N0}.Foreach element t = kΔt ∈ T , deﬁne the semigroup Qt and the co-semigroup Qˆt to be the families of operators

Qt =(QΔt)k, Qˆt =(QˆΔt)k,t∈ T. 62 1 Hyperﬁnite Dirichlet Forms

For each t ∈ T, we may deﬁne approximations A(t) of A and Aˆ(t) of Aˆ by 1 A(t) = I − Qt , t 1 Aˆ(t) = I − Qˆt . t

From A(t) and Aˆ(t),wegettheformsE(t)(u, v)=A(t)u, v = u, Aˆ(t)v.

Let E(·, ·) be the standard part of E(·, ·).ThenE(·, ·) is closed. In addition, (E(·, ·),D(E)) satisﬁes the weak sector condition by Remark 1.4.1. Hence, ◦ (E(·, ·),D(E)) is a coercive closed form on H.Let{Rβ | β ∈ (−∞, 0)} ˆ and {Rβ | β ∈ (−∞, 0)} be the resolvent and co-resolvent of (E(·, ·),D(E)), ˆ respectively. Similarly, let {Tt | t ∈ [0, ∞)} and {Tt | t ∈ [0, ∞)} be the semigroup and co-semigroup of (E(·, ·),D(E)), respectively. For t ∈ (0, ∞), we deﬁne

(t) 1 ◦ E (x, y)= (x − Ttx, y),x,y ∈ H. t

By Albeverio et al. [9], Theorem 3.4 (or referring to Proposition 1.6.4), we have ◦ (t) Lemma 1.7.1. (i) Let x ∈ H.Thenx ∈ D(E) if and only if supt>0 E (x, x) < ∞. (ii) For all x, y ∈ D(E), we have

lim E(t)(x, y)=E(x, y). t↓0

(iii) For all x ∈ D(E), we have

lim E1(x − Ttx, x − Ttx)=0. t↓0

By applying Lemma 1.7.1, we can see from the proof of Theorem 1.6.1 that there exists an inﬁnitesimal δ ∈ T such that (E(·, ·),D(E)) is the standard (δ) δ part of E (u, v) (referring to Remark 1.6.1, and replacing −γRγ by Q in the proof of Theorem 1.6.1). In addition, for any x ∈ ◦H,u ∈ x,andfor ∗ ◦ s all t ∈ [0, ∞),s ∈{kδ | k ∈ N0}, t = s,wehavethatQ u ∈ Ttx.Since s +s s s Q 1 2 ≈ Q 1 if s2 ≈ 0,wehavethatQ u ∈ Ttx for all t ∈ [0, ∞),s ∈ T = ∗ ◦ {kΔt | k ∈ N0}, t = s.Noticethat(E(·, ·),D(E)) is the standard part of both E(u, v) and E(δ)(u, v), and so the resolvent of E(u, v) is almost the same as that of E(δ)(u, v) by Theorem 1.6.2.

Summarizing above results, we have Theorem 1.7.1. Let E(·, ·) be a hyperﬁnite weak coercive quadratic form on a hyperﬁnite dimensional space H. Then, we have 1.7 Weak Coercive Quadratic Forms, Revisited 63

◦ (s) (i) Let u ∈ H.Thenu ∈D(E) if and only if sups E (u, u) < ∞. (ii) For all u, v ∈D(E), we have

lim E(s)(u, v)=E(u, v). ◦s↓0

(iii) For all u ∈D(E), we have

s s lim E1(u − Q u, u − Q u) ≈ 0. ◦s↓0

◦ ∗ ◦ Proof. Let x = u. Then for all t ∈ [0, ∞),s ∈ T = {kΔt | k ∈ N0}, t = s, s we have that Q u ∈ Ttx. By Lemma 1.7.1, the theorem follows easily.

Proposition 1.7.1. Let E(·, ·) be a hyperfinite weak coercive quadratic form on a hyperfinite dimensional linear space H.If◦E(u, u) < ∞, then for all finite s>0,s∈ T, and ◦s =0, we have ◦[Qsu] ∈ D(E).

◦ ◦ s Proof. Let x = u.Fort = s,wehavethatQ u ∈ Ttx. Therefore, we have ◦ s [Q u]=Ttx ∈ D(E).

Chapter 2 Potential Theory of Hyperﬁnite Dirichlet Forms

Probabilistic potential theory has been a very important component in the study of hyperfinite Dirichlet space theory. It provides a probabilistic interpretation of potential theory; and, more generally, it establishes a beautiful bridge between functional analysis and the theory of Markov processes. There are many applications of this theory, especially in the area of infinite dimensional stochastic analysis and mathematical physics. Our purpose in this chapter is to develop the probabilistic potential theory associated with hyperfinite Dirichlet forms and the related Markov chains. The motivation is twofold. On the one hand, we want to establish a relationship between the standard Dirichlet space theory and the hyperfinite counterpart. On the other hand, we want to provide new methods for the theory of hyperfinite Dirichlet forms itself. Infinite dimensional stochastic analysis has been developed extensively in the last decades. We hope to convince the reader that nonstandard analysis can provide a new tool to deal with problems in this exciting area, see particularly Chap. 4, for example.

The arrangement of the present chapter is as follows. In Sect. 2.1,we shall define exceptional sets for non-symmetric hyperfinite Markov chains. Sect. 2.2 will discuss excessive functions and equilibrium potentials. More- over, we introduce a capacity theory for hyperfinite quadratic forms and show that it is a Choquet capacity in Sect. 2.3.Furthermore,weestablisha relation between the family of exceptional sets and the family of zero capacity sets in Sect. 2.4. In Sect. 2.5, we consider positive measures of hyperfinite energy integrals and the associated theory. That is, we establish connections among hyperfinite excessive functions and hyperfinite potentials. Zero capacity subsets will be characterized by positive measures of hyperfinite energy integrals. In Sect. 2.6, we introduce internal additive functionals. The relationship between hyperfinite measures and additive functionals will be considered. Moreover, we shall obtain a positive hyperfinite measure μ associated with an internal function u. In Sect. 2.7, we get Fukushima’s decomposition theorem under individual probability measures. This extends the work of Albeverio et al. [25]andFan[166]. In Sect. 2.8, we shall discuss the properties of internal multiplicative functionals, subordinate semigroups, subprocesses,

S. Albeverio et al., Hyperfinite Dirichlet Forms and Stochastic Processes, 65 Lecture Notes of the Unione Matematica Italiana 10, DOI 10.1007/978-3-642-19659-1_2, c Springer-Verlag Berlin Heidelberg 2011 66 2 Potential Theory and a Feynman-Kac formula. The motivation for this work is given by the corresponding standard theory developed by Blumenthal and Getoor [96]. This serves as a basis for the development of a perturbation theory of hyperfinite Dirichlet forms characterized by internal additive functionals. The content of Sect. 2.9 is the nonstandard version of time change of standard Dirichlet forms. In standard Dirichlet space theory, the question how to change a non- conservative symmetric Markov process into a conservative one has received an answer in Fukushima and Takeda [184] in terms of the Girsanov transformation. In Sect. 2.10, we show that this problem is quite simple in the hyperfinite setting (Theorem 2.10.1).

In this chapter, we shall use the notations developed in Sect. 1.5. However, we need not assume all the conditions required in Sect. 1.5. In Sect. 2.1, we will define exceptional sets for a hyperfinite Markov chain X. We do not need any- thing about the dual hyperfinite Markov chain of X. Starting from Sect. 2.2, we shall work under the setting of Sect. 1.5. That is, we shall assume the conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14) in Sect. 1.5, and the related notations.

2.1 Exceptional Sets

Albeverio et al. [25] has given us a definition of exceptional sets in the framework of the theory of symmetric hyperfinite Dirichlet forms, which is too restrictive for certain cases. In Fan [166], we extended this concept in the same symmetric setting by removing a lot of unnecessary assumptions. In this section, we shall define exceptional sets for non-symmetric hyperfinite Markov chains.

2.1.1 Exceptional Sets

We take an inﬁnitesimal Δt such that Δt > 0.Set

∗ T = {kΔt | k ∈ N0} . (2.1.1)

Let Y be a Hausdorﬀ space and let ∗Y be the nonstandard extension of Y . ∗ ∗ Let S = {s0,s1,...,sN } be an S-dense subset of Y for some N ∈ N − N and let m be a hyperﬁnite measure on S.DenotebyS the internal algebra of subsets of S. Let Q = {qij } be an (N +1)× (N +1) matrix with nonnegative entries. Assume that 2.1 Exceptional Sets 67 N qij =1 for all i =0, 1,...,N, (2.1.2) j=0 and assume that the state s0 is a trap, i.e.,

q0i =0 for all i =0. (2.1.3)

If (Ω,P) is an internal measure space, and X : Ω × T −→ S is an internal process, let

[ω]t = {ω ∈ Ω | X(ω ,s)=X(ω,s) for all s ≤ t}. (2.1.4)

For each t ∈ T, let Ft be the internal algebra on Ω generated by the sets [ω]t.

Assume that for all ω ∈ Ω,wehave

P ([ω]0)=m{X(ω,0)}, (2.1.5) and whenever X(ω,t)=si,wehave

P {ω ∈ [ω]t | X(t + Δt, ω )=sj} = qij P ([ω]t). (2.1.6)

That is, X is a hyperﬁnite Markov chain with the initial distribution m and transition matrix Q. The family (Ω,Ft,Pi,i∈ S) of internal probability spaces is deﬁned by k−1 Pi [ω]kΔt = δiω(0) qω(nΔt),ω((n+1)Δt), (2.1.7) n=0 for each i ∈ S,whereδij is the Kronecker symbol as in Chap. 1.

As in Sect. 1.5,letδ ∈ T and let the sub-line

∗ Tδ = {kδ | k ∈ N0} .

Set

T fin = {t ∈ T | t is finite} , fin Tδ = {t ∈ Tδ | t is finite} .

In addition, for r ∈ T, we let

T r = {t ∈ T | t ≤ r} , r Tδ = {t ∈ Tδ | t ≤ r} .

(δ) Moreover, we know that X is the restriction X|Tδ. 68 2 Potential Theory

For every y ∈ Y, let us deﬁne the monad μ(y) of y by μ(y)= {∗O | O is open such that y ∈ O} .

We call a point y ∈ ∗Y nearstandard if and only if y ∈ μ(x) for some x ∈ Y. Denote by Ns(∗Y ) the set of all nearstandard points in ∗Y .SinceY is a Hausdorﬀ topological space, each element y ∈ Ns(∗Y ) is nearstandard to exactly one element x in Y (we refer to page 48, 2.1.6. Proposition, [25]). We call x the standard part of y and denote it by ◦y or st(y).Inparticular,we can take Y = R and use this notation also.

Deﬁnition 2.1.1. (i) A subset B of S0 is called δ-exceptional for X if ﬁn L(P ) ω∃t ∈ Tδ X(ω,t) ∈ B =0, (2.1.8)

where L(P ) is the Loeb measure of P .

(ii) A subset B of S0 is called exceptional for X if it is δ-exceptional for some inﬁnitesimal δ ∈ T .

Remark 2.1.1. For symmetric hyperfinite Markov chains, δ-exceptional sets are defined in Albeverio et al. [25] in the following way fin L(P ) ω(X(ω,0) ∈ S0) ∧ ∃t ∈ Tδ X(ω,t) ∈ B =0, (2.1.9)

∗ where S0 = S0 ∩ Ns( Y ). Therefore, if a subset B is δ-exceptional in the sense of our Deﬁnition 2.1.1,itisδ-exceptional in the sense of (2.1.9). We have noticed this result in Fan [166] for the symmetric case. Here we deal with the general hyperﬁnite non-symmetric Markov chain X(t).

Remark 2.1.2. From (2.1.8), we see that for every exceptional set B

L(P ) {ω | X(ω,0) ∈ B} =0.

This implies that L(m)(B)=0,whereL(m) is the Loeb measure of m.

We have the following lemma, whose proof is easy and therefore will be omitted.

Lemma 2.1.1. (i) All internal subsets B ⊂ S0 with m(B) ≈ 0 are exceptional.

(ii) The families of exceptional and δ-exceptional sets are closed under countable unions. 2.1 Exceptional Sets 69

Deﬁnition 2.1.2. (i) A δ-exceptional subset A of S0 is called properly δ- exceptional for X if there is a family {Bm,n | m, n ∈ N} of internal subsets such that A = Bm,n m∈N n∈N

and for all si ∈/ A, ﬁn L(Pi) ω∃t ∈ Tδ X(ω,t) ∈ A =0,

where L(Pi) is the Loeb measure of Pi.

(ii) A subset A of S0 is called properly exceptional for X if it is properly δ-exceptional for some δ ≈ 0,δ∈ T.

Proposition 2.1.1. If A ⊂ S0 is a δ-exceptional set, there is a properly δ-exceptional set B ⊃ A.

Proof. Since A is δ-exceptional, there is an internal subset Bmn for each pair (m, n) of natural numbers such that ﬁn 1 A ⊂ Bmn,L(P ) ω∃t ∈ T X(ω,t) ∈ Bmn ≤ . (2.1.10) δ n2m Deﬁne m 1 Cmn = i ∈ SPi ∃t ∈ Tδ X(t) ∈ Bmn ≥ nm and ¯ A = Cmn. m∈N n∈N

Let σ(ω) be a stopping time deﬁned by m 1 σ(ω)=min t ∈ TδPX(ω,t) ∃s ∈ Tδ X(s) ∈ Bmn ≥ . mn

Then, we have fin 1 L(P ) ω∃t ∈ Tδ X(ω,t) ∈ Cmn mn fin m 1 1 = L(P ) ω∃t ∈ Tδ PX(ω,t) ∃s ∈ Tδ X(s) ∈ Bmn ≥ mn mn 70 2 Potential Theory 1 = L(P ) 1(◦σ(ω)<∞) mn m ≤ L(P ) 1(◦σ(ω)<∞)PX(σ(ω)) ∃s ∈ Tδ X(s) ∈ Bmn ◦ m = L(P ) σ(ω) < ∞ ∃s ∈ Tδ X(ω,s + σ(ω)) ∈ Bmn fin ≤ L(P ) ω∃s ∈ Tδ X(ω,s) ∈ Bmn . (2.1.11)

Therefore, we get from the relations (2.1.10)and(2.1.11)that ﬁn 1 L(P ) ω∃t ∈ Tδ X(ω,t) ∈ Cmn ≤ . n

This implies that fin L(P ) ω∃t ∈ Tδ X(ω,t) ∈ Cmn =0. n∈N Therefore, n∈N Cmn is δ-exceptional. By Lemma 2.1.1 (ii), we know that ¯ A = Cmn m∈N n∈N is also δ-exceptional. From the definition of the family {Cmn | n, m ∈ N} , we ¯ see that for any si ∈/ A, fin L(Pi) ω∃t ∈ Tδ X(ω,t) ∈ A =0. (2.1.12)

By using the principle of mathematical induction, we obtain an increasing sequence

A0 ⊂ A1 ⊂ A2 ⊂··· ¯ of δ-exceptional sets, where A0 = A and Ai+1 = Ai for every i. Put B = i∈N Ai.ThenB is a δ-exceptional set of the form (i) B = Cmn i∈N m∈N n∈N

(i) for some internal subsets Cmn. From (2.1.12), we know that if si ∈/ B, then 2.1 Exceptional Sets 71 ﬁn L(Pi) ω∃t ∈ Tδ X(ω,t) ∈ B =0.

We have completed the proof of Proposition 2.1.1.

Remark 2.1.3. The proof of Proposition 2.1.1 is similar to Albeverio et al. [25] 5.4.7 Lemma. For a similar result, see also Fan [166].

2.1.2 Co-Exceptional Sets

ˆ Let Q = {qˆij } be an (N +1)× (N +1) matrix with nonnegative entries. Assume that N qˆij =1 for all i =0, 1,...,N, (2.1.13) j=0 and assume that the state s0 is a trap, i.e.,

qˆ0i =0 for all i =0. (2.1.14)

In the same manner as above, we shall write qˆij for qˆsisj , respectively, whenever it is convenient.

If (Ω,ˆ Pˆ) is an internal measure space, and Xˆ : Ωˆ × T −→ S is an internal process, let

ˆ ˆ ˆ [ˆω]t = {ωˆ ∈ Ω | X(ˆω ,s)=X(ˆω,s) for all s ≤ t}. (2.1.15) ˆ ˆ For each t ∈ T, let Ft be the internal algebra on Ω generated by the sets [ˆω]t.

Assume that for all ωˆ ∈ Ωˆ ˆ ˆ P ([ˆω]0)=m{X(ˆω,0)}, (2.1.16) ˆ and whenever X(ˆω,t)=si,wehave

ˆ ˆ ˆ P{ωˆ ∈ [ˆω]t | X(t + Δt, ωˆ )=sj} =ˆqij P ([ˆω]t). (2.1.17)

That is, Xˆ is a hyperﬁnite Markov chain with initial distribution m and ˆ ˆ ˆ ˆ transition matrix Q. The family (Ω,Ft, Pi,i ∈ S) of internal probability spaces is deﬁned by 72 2 Potential Theory k−1 ˆ Pi [ˆω]kΔt = δiωˆ(0) qˆωˆ(nΔt),ωˆ((n+1)Δt), (2.1.18) n=0

for each i ∈ S.

Similarly as in Sect. 2.1.1, we have the following results about Xˆ. Here, we remark that the dual condition (1.5.13) and the regular condition (1.5.14)of Sect. 1.5 of Chap. 1 are not necessary for the results in this section. ˆ Deﬁnition 2.1.3. (i) A subset B of S0 is called δ-co-exceptional for X if ˆ ﬁn ˆ L(P) ω∃t ∈ Tδ X(ω,t) ∈ B =0,

where L(Pˆ) is the Loeb measure of Pˆ. ˆ (ii) A subset B of S0 is called co-exceptional for X if it is δ-co-exceptional for some inﬁnitesimal δ ∈ T .

Lemma 2.1.2. (i) All internal subsets B ⊂ S0 with m(B) ≈ 0 are co- exceptional.

(ii) The families of co-exceptional and δ-co-exceptional sets are closed under countable unions.

Deﬁnition 2.1.4. (i) A δ-co-exceptional subset A of S0 is called properly δ- ˆ co-exceptional for X if there is a family {Bm,n | m, n ∈ N} of internal subsets such that A = Bm,n m∈N n∈N

and for all si ∈/ A, ˆ ﬁn ˆ L(Pi) ω∃t ∈ Tδ X(ω,t) ∈ A =0,

ˆ ˆ where L(Pi) is the Loeb measure of Pi. ˆ (ii) A subset A of S0 is called properly co-exceptional for X if it is properly δ-co-exceptional for some δ ≈ 0,δ∈ T.

Proposition 2.1.2. If A ⊂ S0 is a δ-co-exceptional set, there is a properly δ-co-exceptional set B ⊃ A. 2.2 Excessive Functions and Equilibrium Potentials 73 2.2 Excessive Functions and Equilibrium Potentials

Let Y be a Hausdorff space and ∗Y be the nonstandard extension of Y .Let ∗ ∗ S = {s0,s1,...,sN } be an S-dense subset of Y for some N ∈ N−N and let m be a hyperfinite measure on S.DenotebyS the internal algebra of subsets of S. Let X be a hyperfinite Markov chain with the initial distribution m and transition matrix Q, which is introduced in Sect. 2.1 by the conditions (2.1.1), (2.1.2), (2.1.3), (2.1.4), (2.1.5), (2.1.6), and (2.1.7). Moreover, let Xˆ be a hyperfinite Markov chain with the initial distribution m and transition matrix Qˆ, which is introduced in Sect. 2.1 by the conditions (2.1.13), (2.1.14), (2.1.15), (2.1.16), (2.1.17), and (2.1.18).

We assume the measure m and the transition matrices Q and Qˆ satisfy the dual conditions

miqij = mjqˆji for all i =0,j =0. (2.2.1)

Besides, we assume that

mi =0 for at least one i =0. (2.2.2)

Hence X and Xˆ are dual hyperﬁnite Markov chains.

We have deﬁned an inner product ·, · on the hyperﬁnite dimensional space H by uv dm = u, v S0 N = u(si)v(si)m(si). (2.2.3) i=1

Let ||·||be the norm generated by ·, ·.Let{Qt | t ∈ T }, A and E(·, ·) be the semigroup, infinitesimal generator and hyperfinite quadratic form of X. Similarly, let {Qˆt | t ∈ T }, Aˆ,andEˆ(·, ·) be the co-semigroup, infinitesimal co-generator, and hyperfinite quadratic co-form of Xˆ. We remark here that the reader may refer to Sect. 1.5 of Chap. 1 for the other notations.

Besides, we have for α ∈ ∗R,α≥ 0 and δ ∈ T that

(δ) (δ) Eα (u, v)=E (u, v)+αu, v, ˆ(δ) ˆ(δ) Eα (u, v)=E (u, v)+αu, v.

Each of these forms generates a norm (possibly a semi-norm in the case α =0): 74 2 Potential Theory

1 (δ) (δ) 2 |u|α = Eα (u, u)

1 ˆ(δ) 2 = Eα (u, u) .

Deﬁnition 2.2.1. Fix α ∈ ∗R,α≥ 0 and δ ∈ T. (i) An element u ∈ H is called hyperﬁnite α-excessive associated with E(δ)(·, ·), if

u(i) ≥ 0,Qδu(i) ≤ (1 + αδ)u(i)

for every i ∈ S0. (ii) An element u ∈ H is called hyperﬁnite α-co-excessive associated with Eˆ(δ)(·, ·), if

u(i) ≥ 0, Qˆδu(i) ≤ (1 + αδ)u(i)

for every i ∈ S0.

Lemma 2.2.1. The following results hold: (i) Let u ∈ H be hyperﬁnite α-excessive associated with E(δ)(·, ·). Then

Qkδu(i) ≤ (1 + αδ)ku(i), ∀k ∈ ∗N, (2.2.4)

for every i ∈ S0. (ii) Let u ∈ H be hyperﬁnite α-co-excessive associated with Eˆ(δ)(·, ·). Then

Qˆkδu(i) ≤ (1 + αδ)ku(i), ∀k ∈ ∗N, (2.2.5)

for every i ∈ S0.

Proof. (i) We have from the Markov property of X that

2δ Q u(i)=Eiu(X(2δ))

= EiEX(δ)u(X(δ)) ≤ (1 + αδ)Eiu(X(δ)) ≤ (1 + αδ)2u(i).

Now it is easy to prove (2.2.4) by mathematical induction.

(ii) In the same way as above, we get (2.2.5).

Remark 2.2.1. The results of Lemma 2.2.1 are true for all hyperﬁnite Markov chains X and Xˆ without dual condition (2.2.1) and regular condition (2.2.2). 2.2 Excessive Functions and Equilibrium Potentials 75

Let A be an internal subset of S0,andletδ ∈ T. Deﬁne a stopping time (δ) σA by (δ) (δ) σA (ω)=min t ∈ Tδ | X (ω,t) ∈ A .

Let f : A −→ ∗R be an internal function deﬁned on A.Forα ∈ ∗R,α ≥ 0, set (δ) (δ) (δ) −σA /δ (δ) eα (f)(i)=Ei (1 + αδ) f X (σA ) .

(δ) Similarly, we define a stopping time σÂ by (δ) ˆ (δ) σÂ (ˆω)=min t ∈ Tδ | X (ˆω,t) ∈ A .

For α ∈ ∗R,α≥ 0, set (δ) (δ) (δ) ˆ −σˆA /δ ˆ (δ) eˆα (f)(i)=Ei (1 + αδ) f X (ˆσA ) .

Then, we have Proposition 2.2.1. Assume that E(·, ·) and Eˆ(·, ·) are the hyperfinite quadratic form and co-form associated with the dual hyperfinite Markov ˆ ∗ chains X and X, respectively. Let f : A −→ R be an internal function (δ) (δ) defined on an internal subset A of S0.ThenEα eα (f),u =0and (δ) (δ) Eα u, eˆα (f) =0for all u which are zero on A.

Proof. We notice that if i/∈ A,then

N (δ) (δ) (δ) (1 + αδ) eα (f)(i)= eα (f)(j)qij . j=1

If u is zero on A,then ⎡ ⎤ 1 N N (δ) (δ) ⎣ (δ) (δ) (δ)⎦ Eα eα (f),u = eα (f)(i) − eα (f)(j)qij u(i)mi δ i=1 j=1 N (δ) +α eα (f)(i)u(i)mi i=1⎡ ⎤ 1 N N ⎣ (δ) (δ) (δ)⎦ = (1 + αδ)eα (f)(i) − eα (f)(j)qij u(i)mi δ i=1 j=1 =0, 76 2 Potential Theory since the expression in the bracket is zero when i/∈ A,andu(i)=0when (δ) (δ) i ∈ A. Similarly, we can prove Eα u, eˆα (f) =0. Hence, Proposition 2.2.1 is proved.

Corollary 2.2.1. Assume that E(·, ·) and Eˆ(·, ·) are the hyperfinite quadratic form and co-form associated with the dual hyperfinite Markov chains X and Xˆ, respectively. Let f : A −→ ∗R be an internal function defined on an internal subset A of S0.Set ∗ M(f)= gg : S0 −→ R is internal and g |A= f .

Then, we have for all g ∈M(f) (δ) (δ) (δ) (δ) (δ) Eα eα (f),eα (f) = Eα eα (f),g (δ) (δ) = Eα g,eˆα (f) (δ) (δ) (δ) = Eα eˆα (f), eˆα (f) . (2.2.6)

Proof. It follows from Proposition 2.2.1 that for all g ∈M(f) (δ) (δ) (δ) (δ) (δ) Eα eα (f),eα (f) = Eα eα (f),g (2.2.7) and (δ) (δ) (δ) (δ) (δ) Eα g,eˆα (f) = Eα eˆα (f), eˆα (f) . (2.2.8)

(δ) (δ) Since eα (f), eˆα (f) ∈M(f), we have from the relations (2.2.7)and(2.2.8) that (δ) (δ) (δ) (δ) (δ) (δ) Eα eα (f),eα (f) = Eα eα (f), eˆα (f) (δ) (δ) (δ) = Eα eˆα (f), eˆα (f) .

Hence, (2.2.6)isproved.

Corollary 2.2.2. Assume that E(·, ·) is a symmetric hyperﬁnite quadratic form, i.e., E(·, ·)=Eˆ(·, ·).Letf : A −→ ∗R be an internal function deﬁned on an internal subset A of S0. Then, we have (δ) (δ) (δ) (δ) Eα eα (f),eα (f) =min Eα (g,g) | g ∈M(f) . 2.2 Excessive Functions and Equilibrium Potentials 77

(δ) (δ) Proof. First, we observe that eα (f)=ˆeα (f) satisﬁes (δ) (δ) (δ) (δ) Eα eα (f),u = Eα u, eα (f) =0 (2.2.9)

for all u which are zero on A. In fact, this is implied by Proposition 2.2.1 and the symmetry of E(·, ·).

To prove our corollary, let w be another extension of f, i.e., w ∈M(f). Then u = w − eα(f) is zero on A,andby(2.2.9) (δ) (δ) (δ) (δ) Eα (w, w)=Eα eα (f)+u, eα (f)+u (δ) (δ) (δ) (δ) (δ) (δ) = Eα eα (f),eα (f) +2Eα eα (f),u + Eα (u, u) (δ) (δ) (δ) (δ) = Eα eα (f),eα (f) + Eα (u, u).

Deﬁnition 2.2.2. Let f : A −→ ∗R be an internal function deﬁned on (δ) an internal subset A of S0. The function eα (f) is called the equilibrium (δ) α-potential of f,andeˆα (f) is called the co-equilibrium α-potential of f associated with E(δ)(·, ·).

Lemma 2.2.2. Let A be an internal subset of S0 and 1A be the indicator ∗ function of A. Then for all α ∈ R+ and δ ∈ T, we have (δ) (i) eα (1A)(i) is hyperﬁnite α-excessive. (δ) (ii) eˆα (1A)(i) is hyperﬁnite α-co-excessive. Proof. (i) First of all, we notice that if i/∈ A,

N (δ) (δ) (δ) (1 + αδ) eα (1A)(i)= eα (1A)(j)qij j=1 δ (δ) = Q eα (1A)(i). (2.2.10)

For i ∈ A,wehave

δ (δ) Q eα (1A)(i) ≤ 1 (δ) = eα (1A)(i) (δ) ≤ (1 + αδ)eα (1A)(i). (2.2.11)

From the relations (2.2.10)and(2.2.11), we have got (i).

(ii) The proof of this is similar to the one for (i). 78 2 Potential Theory

Theorem 2.2.1. Assume that E(·, ·) and Eˆ(·, ·) are the hyperﬁnite quadratic form and co-form associated with the dual hyperﬁnite Markov chains X and ˆ X, respectively. Let A be an internal subset of S0 and let 1A be the indicator function of A.Set ∗ L(1A)= gg : S0 −→ R is internal and g(i) ≥ 1, ∀i ∈ A .

∗ Then for all α ∈ R+,δ ∈ T,u ∈L(1A), we have (δ) (δ) (δ) (δ) (δ) Eα eα (1A),u ≥Eα eα (1A),eα (1A) , (δ) (δ) (δ) (δ) (δ) Eα u, eˆα (1A) ≥Eα eˆα (1A), eˆα (1A) .

Proof. In the same way as the proof of Proposition 2.2.1, we get our theorem by Lemma 2.2.2.

Corollary 2.2.3. Assume that E(δ)(·, ·) is a hyperﬁnite weak coercive quadratic form with continuity constant C.LetA be an internal subset of S0.Then

(1) For all u ∈L(1A), we have (δ) (δ) (δ) 2 (δ) E1 e1 (1A),e1 (1A) ≤ C E1 (u, u), (δ) (δ) (δ) 2 (δ) E1 eˆ1 (1A), eˆ1 (1A) ≤ C E1 (u, u).

(2) If C =1, we have (δ) (δ) (δ) (δ) (δ) (δ) E1 e1 (1A),e1 (1A) = E1 eˆ1 (1A), eˆ1 (1A) (δ) =min{E1 (u, u)u ∈L(1A)}.

Proof. The proof is very easy and will therefore be omitted!

2.3 Capacity Theory

In this section, we assume that all conditions in Sect. 2.2 are satisfied, and we shall use the same notations there. That is, X and Xˆ are dual hyperfinite Markov chains, and E(·, ·) and Eˆ(·, ·) are the hyperfinite quadratic form and co-form associated with X and Xˆ, respectively. Let A be an internal subset ∗ of S0.Forα ∈ R+,δ ∈ T, set 2.3 Capacity Theory 79 (δ) (δ) (δ) (δ) Capα (A)=Eα eα (1A),eα (1A) .

(δ) (δ) We call Capα (A) the α-capacity of A associated with E (·, ·). An internal (δ) ◦ (δ) (δ) (δ) subset A has ﬁnite energy of E (·, ·) if E1 e1 (1A),e1 (1A) < ∞. We (δ) (δ) (δ) shall write eα (A) for eα (1A). Moreover, we call eα (A) the equilibrium α-potential of A associated with E(δ)(·, ·). Whenever δ = Δt, we abbreviate (δ) (δ) eα (f) and Capα (f) by eα(f) and Capα(f), respectively.

By Corollary 2.2.1,wehave (δ) (δ) (δ) Eα eα (A),eα (A) = Eα eα(A), 1 .

(δ) (δ) Applying the Beurling–Deny formulae, Lemma 1.5.1,toEα (eα (A), 1), we get

N N (δ) (δ) (δ) (δ) (δ) qˆi0 Eα eα (A),eα (A) = α eα (A)(i)mi + eα (A)(i) mi. i=1 i=1 δ (2.3.1)

The equilibrium potentials will serve as bridgeheads in our campaign to unify the analytic theory of Dirichlet forms and the probabilistic theory of Markov processes. The key to their importance is the observation that they have natural interpretations both in analytic and in probabilistic terms. On the (δ) one hand, they minimize the forms Eα (·, ·) under suitable side conditions. On the other they describe how the process X hits subsets of S0. For a fuller understanding of the probabilistic description, we point out that for ﬁnite α

◦ (δ) ◦ −ασ(δ) eα (A)(i)=Ei( e A ),

◦ −ασ(δ) (δ) and that the function α → Ei( e A ) is the Laplace transform of σA . Hence, the distribution of σA can be completely recovered from the functions (δ) eα (A). This fact will be of great importance in Chap. 3.

Lemma 2.3.1. Let δ ∈ T,α ≥ 0 and α ∈ ∗R, then the following results hold:

(i) If A and B are two internal subsets of S0, A ⊂ B, then we have

(δ) (δ) Capα (A) ≤ Capα (B).

(ii) If A and B are internal subsets of S0, then we have

(δ) (δ) (δ) (δ) Capα (A ∪ B)+Capα (A ∩ B) ≤ Capα (A)+Capα (B). 80 2 Potential Theory

(δ) (δ) Proof. (i) Since eα (B)−eα (A) ≥ 0 on B, we have from Corollary 2.2.1 and Theorem 2.2.1 (δ) (δ) (δ) (δ) Capα (A)=Eα eα (A), eˆα (A) (δ) (δ) (δ) = Eα eα (A), eˆα (B) (δ) (δ) (δ) ≤Eα eα (B), eˆα (B) (δ) =Capα (B).

(ii) Since

(δ) (δ) (δ) (δ) eα (A)+eα (B) − eα (A ∪ B) − eα (A ∩ B) ≥ 0 on A ∪ B,

we have the following from Theorem 2.2.1 (δ) (δ) (δ) (δ) Capα (A)+Capα (B) − Capα (A ∪ B) − Capα (A B) (δ) (δ) (δ) (δ) (δ) (δ) = Eα eα (A)+eα (B) − eα (A ∪ B) − eα (A ∩ B), eˆα (A ∪ B) ≥ 0.

Now let us generalize our capacity theory. For every subset A of S0,we deﬁne (δ) (δ) Capα (A)=inf Capα (B)A ⊂ B,B ∈S0 .

(δ) (δ) We call Capα (A) the α-capacity of A associated with E (·, ·).

Lemma 2.3.2. Let δ ∈ T,α ≥ 0 and α ∈ ∗R, then the following results hold:

(i) If {An | n ∈ N} is an increasing sequence of internal subsets of S0,then we have ◦ (δ) ◦ (δ) Capα An =sup Capα (An)n ∈ N . n∈N

(ii) If {An | n ∈ N} is a decreasing sequence of internal subsets of S0,then we have ◦ (δ) ◦ (δ) Capα An =inf Capα (An)n ∈ N . n∈N 2.3 Capacity Theory 81 ◦ (δ) Proof. (i) Set a = sup Capα (An)n ∈ N . Obviously, we have ◦ (δ) a ≤ Capα An . n∈N

∗ Hence, we can suppose that a<∞. Let {An | n ∈ N} be an increasing internal extension of {An | n ∈ N}.Givenε>0, we consider the following internal set n ∗ n ∈ N Al = An is internal, l=1 n (δ) (δ) Capα Al =Capα (An) ≤ a + ε . l=1

It is easy to see that N is contained in above internal set. By saturation, there is an inﬁnite member M belonging to it. Therefore, we have (δ) (δ) Capα An ≤ Capα (AM ) n∈N ≤ a + ε. ◦ (δ) By letting ε ↓ 0,weget Capα n∈N An ≤ a. Therefore, we have ◦ (δ) ◦ (δ) Capα An =sup Capα (An)n ∈ N . n∈N

∗ (ii) We extend{An | n ∈ N} to be a decreasing sequence {An | n ∈ N} . ◦ (δ) When inf Capα (Al)l ∈ N < ∞, consider the following internal subset for ε>0 ∗ (δ) ◦ (δ) n ∈ NCapα (An) ≥ inf Capα (Al)l ∈ N − ε .

By saturation, there is an inﬁnite M belonging to the above internal set. Hence, we have ∞ ◦ (δ) ◦ (δ) Capα Al ≥ Capα (AM ) l=1 ◦ (δ) ≥ inf Capα (Al)l ∈ N − ε. 82 2 Potential Theory

By letting ε ↓ 0,weget ∞ ◦ (δ) ◦ (δ) Capα Al ≥ inf Capα (Al)l ∈ N . l=1

On the other hand, it is easy to see that ∞ ◦ (δ) ◦ (δ) Capα Al ≤ inf Capα (Al) . l∈N l=1 ◦ (δ) If inf Capα (Al)l ∈ N = ∞, we consider the following internal subset for N0 ∈ N ∗ (δ) n ∈ NCapα (An) ≥ N0 .

By saturation and letting N0 ↑∞, we can show that (ii) holds.

Lemma 2.3.3. If {An | n ∈ N} is a sequence of internal subsets of S0,then we have ◦ (δ) ◦ (δ) Capα An ≤ Capα (An), (2.3.2) n∈N n∈N for all δ ∈ T,α ≥ 0.

◦ (δ) Proof. Set b = n∈N Capα (An). If b = ∞, the inequality (2.3.2)holds.In ∗ the following proof, we shall assume b<∞.Let{An | n ∈ N} be an internal extension of {An | n ∈ N} . For every ε>0, it follows from Lemma 2.3.1 (ii) that n n (δ) (δ) Capα Al ≤ Capα (Al) ≤ ε + b for all n ∈ N. l=1 l=1

Consider the following internal set n n n ∗ (δ) (δ) n ∈ N Al is internal and Capα Al ≤ Capα (Al) ≤ b + ε . l=1 l=1 l=1

By saturation, there is an inﬁnite element M = M(ε) belonging to the above internal set. Hence, we obtain 2.3 Capacity Theory 83 M (δ) (δ) Capα Al ≤ Capα Al l∈N l=1 M (δ) ≤ Capα (Al) l=1 ≤ b + ε.

By letting ε ↓ 0, we have proved the inequality (2.3.2). Proposition 2.3.1. For all δ ∈ T, α ≥ 0, we have

(i) If A and B are two subsets of S0, A ⊂ B, then

(δ) (δ) Capα (A) ≤ Capα (B) (2.3.3)

(ii) Let {An | n ∈ N} be a sequence of subsets of S0,then ◦ (δ) ◦ (δ) Capα An ≤ Capα (An). (2.3.4) n∈N n∈N

(iii) Let {An | n ∈ N} be an increasing sequence of subsets of S0,then ◦ (δ) ◦ (δ) Capα An =sup Capα (An)n ∈ N . n∈N

Proof. (i) The proof is immediate, using the deﬁnition.

◦ (δ) (ii) Set b = n∈N Capα (An). We can assume that b<∞.Givenε>0, for every n ∈ N, let us take an internal subset Bn such that An ⊂ Bn and

(δ) (δ) Capα (An) ≤ Capα (Bn) ≤ Cap(δ)( )+ ε α An 2n+1 .

Therefore, we have from (i) and Lemma 2.3.3 that ◦ (δ) ◦ (δ) Capα An ≤ Capα Bn n∈N n∈N ◦ (δ) ≤ Capα (Bn) n∈N ◦ (δ) ≤ Capα (An)+ε. n∈N

By letting ε ↓ 0, we get the inequality (2.3.4). 84 2 Potential Theory

◦ (δ) (iii) We may assume that for all n ∈ N, Capα (An) < ∞. Given ε>0, for each n ∈ N,letBn be an internal subset of S0 such that

(δ) (δ) An ⊂ Bn, Capα (Bn) ≤ Capα (An)+ε.

Then, we have from (2.3.3) and Lemma 2.3.2 (i) that ◦ (δ) ◦ (δ) Capα An ≤ Capα Bn n∈N n∈N ◦ (δ) =sup Capα (Bn) n ◦ (δ) ≤ sup Capα (An)+ε n ◦ (δ) ≤ sup Capα (An) + ε. n

By letting ε ↓ 0,weget ◦ (δ) ◦ (δ) Capα An ≤ sup Capα (An)n ∈ N . (2.3.5) n∈N

On the other hand, it is easy to see that ◦ (δ) ◦ (δ) Capα An ≥ sup Capα (An)n ∈ N . (2.3.6) n∈N

From the inequalities (2.3.5)and(2.3.6), we have proved Proposition 2.3.1 (iii).

For the purpose of explaining our Theorem 2.3.1 in the following, we ﬁrst introduce some notations in capacity theory (referring to, e.g., [282]). Let G be a set, G be a family of some subsets of G.DenotebyGσ (respectively, Gδ) the closure of a collection of subsets of G under countable union (respectively, countable intersection). That is, ∞ ∞ Gσ = AnAn ∈G , Gδ = AnAn ∈G . n=1 n=1

Moreover, we shall write Gσδ =(Gσ)δ.

Deﬁnition 2.3.1. Let G be a set. A paving G on G is a family of subsets of G such that the empty set ∅ is contained in G. The pair (G, G) consisting of asetG and a paving G on G is called a paved set. 2.3 Capacity Theory 85

Deﬁnition 2.3.2. Let (G, G) be a paved set. The paving G is said to be semi-compact if every countable family of elements of G, which has the ﬁnite intersection property, has a nonempty intersection.

It is easy to see that (S0, S0) is a semi-compact paved set. Moreover, S0 is closed under the complement, ﬁnite union, and ﬁnite intersection operations.

Deﬁnition 2.3.3. A subset A of S0 is said to be S0-analytic if there exists an auxiliary set G with a semi-compact paving G, and a subset B ⊂ G × S0 belonging to (G×S0)σδ such that A is the projection of B on S0.Wedenote by A(S0) all the S0-analytic sets (we notice that G×S0 = {G1 × S1 | G1 ∈G and S1 ∈S0}).

Lemma 2.3.4. The σ-ﬁeld σ(S0) generated by S0 is contained in A(S0).

Proof. For every F ∈S0, S0 −F belongs to S0 also. By Meyer [282], Chap. III T12 Theorem, we know σ(S0) ⊂A(S0). Definition 2.3.4. An extended real valued set function I :2S0 → S [−∞, +∞], defined on all subsets 2 0 of S0, is called a Choquet S0-capacity if it satisfies the following properties: (i) I is increasing, i.e., A ⊂ B =⇒ I(A) ≤ I(B). (ii) For every increasing sequence {An | n ∈ N} of subsets of S0,wehave I An =supI(An). n∈N n∈N

(iii) For every decreasing sequence {An | n ∈ N} of elements of S0,wehave I An =infI(An). n∈N n∈N

We have reached one of our main results. Theorem 2.3.1. For each δ ∈ T and α ≥ 0,α ∈ ∗R, we have the following results: ◦ (δ) (i) Capα (·) is a Choquet S0-capacity. ◦ (δ) (ii) Every S0-analytic set is capacitable with respect to capacity Capα (·). That is, for every A ∈A(S0), we have ◦ (δ) ◦ (δ) Capα (A)=sup Capα (B)B = Bm,Bm ∈S0 and B ⊂ A . m∈N

(iii) Every subset A of S0 belonging to σ(S0) is capacitable with respect to ◦ (δ) the capacity Capα (·) whenever 0 < st(α) < ∞. 86 2 Potential Theory

◦ (δ) Proof. By Lemma 2.3.1 and Proposition 2.3.1, we know that Capα (·) is aChoquetS0-capacity. Therefore, (ii) holds by Meyer [282], Chap. III T19 Theorem. (iii) is the consequence of (ii) and Lemma 2.3.4.

Deﬁnition 2.3.5. (i) A subset B of S0 is said to be of δ-zero capacity,if (δ) Cap1 (B) ≈ 0. (δ) (ii) A subset B of S0 is said to be of zero capacity if Cap1 (B) ≈ 0 for some inﬁnitesimal δ ∈ T. (δ) Remark 2.3.1. For any B ∈S0 and δ ∈ T, we have m(B) ≤ Cap1 (B). Therefore, for any zero capacity subset B of S0,wehaveL(m)(B)=0.

2.4 Relation of Exceptionality and Capacity Theory

In regular Dirichlet space theory, we know that the concepts of exceptional sets and zero capacity sets are equivalent, see Fukushima [175], Theorem 4.3.1. As the fourth section of this chapter, we will discuss the corresponding problem in our hyperﬁnite Dirichlet space theory.

We shall continue the discussion of Sect. 2.3. Hence, we assume that all conditions in Sect. 2.2 are satisfied in this section as well, i.e., X and Xˆ are dual hyperfinite Markov chains, and E(·, ·) and Eˆ(·, ·) are the hyperfinite quadratic form and co-form associated with X and Xˆ, respectively. Let H be the hyperfinite dimensional space with an inner product ·, · defined by (2.2.3) in Sect. 2.2 or (1.5.15) in Sect. 1.5, Chap. 1. Lemma 2.4.1. { | ∈ N} Let Bn n be a sequence of internal subsets of S0.If ◦ (δ) n ∞ limn→∞ Cap1 ( m=1 Bm)=0, then n=1 Bn is a δ-exceptional set, where δ ∈ T,δ ≈ 0.

Proof. Since S0 is closed under ﬁnite intersection, we may assume that {Bn | n ∈ N} is a decreasing sequence. Deﬁne a stopping time for each n ∈ N,

(δ) ( )=min{ ∈ | ( ) ∈ } σBn ω t Tδ X ω,t Bn .

We have 1 L(P ) ω∃t ∈ Tδ X(ω,t) ∈ Bn ◦ 1 = P ω∃t ∈ Tδ X(ω,t) ∈ Bn ◦ 1 = Ei ω∃t ∈ Tδ X(ω,t) ∈ Bn dm(i) S0 2.4 Relation of Exceptionality and Capacity Theory 87 ◦ = 1 ( ) Ei (σ(δ) ≤1) dm i Bn S0 ◦ (δ) 1 −σB /δ − δ = Ei ω (1 + δ) n ≥ (1 + δ) dm(i) S0 ⎧ ⎫ ◦ ⎨ −σ(δ) /δ ⎬ (1 + δ) Bn ≤ Ei 1 dm(i) ⎩ − δ ⎭ S0 (1 + δ) ◦ −σ(δ) /δ = e · Ei (1 + δ) Bn dm(i) S0 ◦ (δ) = e · e1 (Bn)(i) dm(i) S0 ◦ (δ) (δ) ≤ e · E1 e1 (Bn),e1 (Bn)

◦ (δ) = e · Cap1 (Bn) −→ 0, (2.4.1) where the last inequality comes from (2.3.1) in Sect. 2.3. Then, we have ∞ 1 L(P ) ω∃t ∈ Tδ X(ω,t) ∈ Bn =0. (2.4.2) n=1

By the dual property of the Markov process X(t) and (2.4.2), we also have ∞ ﬁn L(P ) ω∃t ∈ Tδ X(ω,t) ∈ Bn =0. n=1

∞ Therefore, the set n=1 Bn is δ-exceptional.

Theorem 2.4.1. If a subset A of S0 is of δ-zero capacity, it is δ-exceptional.

(δ) Proof. Since Cap1 (A) ≈ 0, we can take a sequence of internal subsets {Bn | n ∈ N} satisfying

∞ n ◦ (δ) A ⊂ Bn, lim Cap ( Bm)=0. n→∞ 1 n=1 m=1

∞ Using Lemma 2.4.1, we know that n=1 Bn is δ-exceptional. Hence, A is δ-exceptional also. This completes the proof of Theorem 2.4.1.

Lemma 2.4.2. Let δ1 ∈ T,δ1 ≈ 0. Assume that for all f ∈ H,if ◦ (δ1) (E1 (f,f)) < ∞ and f(s) ≈ 0 for all s/∈ B, where B is a δ1-exceptional set, (δ1) then we have E1 (f,f) ≈ 0. Let A be a subset of S0.IfA is δ1-exceptional and there exists an internal subset B of S0 such that 88 2 Potential Theory

◦ (δ1) A ⊂ B, Cap1 (B) < ∞, (2.4.3) then A is of δ1-zero capacity.

Proof. By using Proposition 2.1.1, there exists a properly δ1-exceptional set Bm,n ⊃ A. m∈N n∈N

For simplicity, we assume that Bm,n ⊂ B for all n, m ∈ N,andforeachm, the sequence {Bm,n | n ∈ N} is decreasing with respect to n.Inordertoshow that A has zero capacity, we ﬁrst prove that n∈N Bm,n has zero capacity for every m.Fromnowon,weﬁxanm ∈ N.

◦ (δ1) By the assumption (2.4.3), we know that Cap1 (Bm,n) < ∞ for every n. (δ1) ∗ Moreover, Cap1 (Bm,n) is decreasing with n.Let{Bm,n | n ∈ N} be a decreasing extension of {Bm,n | n ∈ N} . By saturation, there exists an inﬁnite ∗ element nm ∈ N − N such that

◦ (δ1) (δ1) (δ1) lim [E (e (Bm,n),e (Bm,n))] n→∞ 1 1 1 = ◦[E(δ1)( (δ1)( ) (δ1)( ))] 1 e1 Bm,nm ,e1 Bm,nm .

Therefore, we have

◦ ◦ E(δ1) (δ1)( ) (δ1)( ) = Cap(δ1)( ) 1 e1 Bm,nm ,e1 Bm,nm 1 Bm,nm < ∞. (δ1) Besides, for every i ∈ S0, it is easy to see that e (Bm,n)(i) | n ∈ N 1 is decreasing with respect to n.Since m∈N n∈N Bm,n is properly δ1- exceptional, we can show (δ1)( )( ) ≈ 0 ∈ e1 Bm,nm i for every i/ Bm,n. (2.4.4) m∈N n∈N

In fact, for every M0 ∈ [0, ∞),wehave (δ1 ) (δ1) −σm,n/δ1 e1 (Bm,n)(i)=Ei (1 + δ1) −σ(δ1 )/δ = (1 + ) m,n 1 1 +1 Ei δ1 (σBm,n ≥M0) (σm,n

−M /δ By letting M0 be suﬃciently large, we know that (1 + δ1) 0 1 will be very small. Taking n suﬃciently large, we see that the approximation (2.4.4)holds. 2.4 Relation of Exceptionality and Capacity Theory 89

The assumption in the Lemma implies that E(δ1) (δ1)( ) (δ1)( ) ≈ 0 1 e1 Bm,nm ,e1 Bm,nm .

Therefore, we get

◦ (δ1) (δ1) (δ1) E1 e1 (Bm,n),e1 (Bm,n) −→ 0.

This says that (δ1) Cap1 Bm,n ≈ 0. (2.4.5) n∈N

By Proposition 2.3.1 (i) and (ii) and the approximation (2.4.5), we obtain ◦ (δ1) ◦ (δ1) Cap1 (A) ≤ Cap1 Bm,n m∈N n∈N ◦ (δ1) ≤ Cap1 Bm,n m∈N n∈N =0.

Thus, the set A has a δ1-zero capacity.

◦ (δ1) Theorem 2.4.2. Let δ1 ∈ T,δ1 ≈ 0. Assume for all f ∈ H,if (E1 (f,f)) < ∞ and f(s) ≈ 0 for all s/∈ B, where B is a δ1-exceptional set, then we (δ1) have E1 (f,f) ≈ 0. Let A be a subset of S0.IfA is δ1-exceptional and there exists a sequence of internal subsets {Bn | n ∈ N} of S0 such that ◦ (δ1) A ⊂ Bn and Cap1 (Bn) < ∞, ∀n ∈ N, n∈N then A is of δ1-zero capacity.

Proof. The proof follows easily from Lemma 2.4.2 and Proposition 2.3.1 (ii).

In Lemma 2.4.2 and Theorem 2.4.2, we talk about the hyperfinite quadratic ◦ (δ1) form and co-form and make one assumption: for all f ∈ H,if (E1 (f,f)) < ∞ and f(s) ≈ 0 for all s/∈ B, where B is a δ1-exceptional set, then we have (δ1) E1 (f,f) ≈ 0. This assumption is somewhat equivalent to say that for all ◦ (δ1) (δ) f,if (E1 (f,f)) < ∞,thenf ∈D(E ). The assumption, however, is not always easy to verify. In the following, we will give results for hyperfinite 90 2 Potential Theory weak coercive quadratic forms, for which we do not make the assumption. The results were first proved for hyperfinite symmetric Dirichlet forms in Fan [166]. The following results extend those of Fan [166].

Lemma 2.4.3. Assume that E(δ)(·, ·) is a hyperfinite weak coercive quadratic form on the space H for all infinitesimal δ ∈ T. Let δ1 ∈ T,δ1 ≈ 0. Let A be asubsetofS0.IfA is δ1-exceptional and there exists an internal subset B of S0 which satisfies condition (2.4.3), then there is an infinitesimal δ0 ∈ T which is larger than δ1 such that A is of δ-zero capacity for all δ ≥ δ0,δ ≈ 0.

Proof. In the proof of Lemma 2.4.2,wehave

◦ ◦ E(δ1) (δ1)( ) (δ1)( ) = Cap(δ1)( ) 1 e1 Bm,nm ,e1 Bm,nm 1 Bm,nm < ∞.

By Corollary 1.2.4 and Theorem 1.4.2,thereisaδm ≈ 0 such that (δ1)( ) ∈D(E(δ)) ≥ e1 Bm,nm for all inﬁnitesimal δ δm. Moreover, we know from the proof of Lemma 2.4.2 that (δ1)( )( ) ≈ 0 ∈ e1 Bm,nm i for every i/ Bm,n. m∈N n∈N

(δ1)( ) ∈D(E(δ)) ≥ ≈ 0 (δ1)( ) 2 Since e1 Bm,nm for all δ δm,δ ,e1 Bm,nm is S - integrable in the sense of Albeverio et al. [25], page 77, Chap. 3. Hence, we (δ1)( ) ≈ 0 have e1 Bm,nm in the hyperﬁnite dimensional space H because L(m) Bm,n =0. m∈N m∈N

E(δ)( (δ1)( ) (δ1)( )) ≈ 0 ≥ ≈ Therefore, we have 1 e1 Bm,nm ,e1 Bm,nm for all δ δm,δ 0. This implies that

◦ (δ) (δ) (δ) E1 e1 (Bm,n),e1 (Bm,n) ↓ 0,n−→ ∞ , for all δ ≥ δm,δ ≈ 0.

Hence, we have (δ) Cap1 Bm,n ≈ 0 for all δ ≥ δm,δ ≈ 0. (2.4.6) n∈N

By saturation, there is a δ0 ≈ 0 larger than all δm,m ∈ N. Therefore, it follows from the approximation (2.4.6)thatforδ ≥ δ0,δ ≈ 0, 2.5 Measures of Hyperﬁnite Energy Integrals 91 (δ) Cap1 Bm,n ≈ 0 for all m ∈ N. (2.4.7) n∈N

By Proposition 2.3.1 (i) and (ii) and the approximation (2.4.7), we obtain ◦ (δ) ◦ (δ) Cap1 (A) ≤ Cap1 Bm,n m∈N n∈N ∞ ◦ (δ) ≤ Cap1 Bm,n m=1 n∈N =0.

Therefore, the set A has δ-zero capacity. Theorem 2.4.3. Let E(δ)(·, ·) be a hyperfinite weak coercive quadratic form on the space H for all infinitesimal δ ∈ T. Let δ1 ∈ T,δ1 ≈ 0. Let A be a subset of S0.IfA is δ1-exceptional and there exists a sequence of internal subsets {Bn | n ∈ N} of S0 such that ◦ (δ1) A ⊂ Bn and Cap1 (Bn) < ∞, ∀n ∈ N, n∈N then there is an infinitesimal δ0 ∈ T which is larger than δ1 such that A is of δ-zero capacity for all δ ≥ δ0,δ ≈ 0. Proof. The proof follows easily from Lemma 2.4.3,Proposition2.3.1 (ii), and saturation.

2.5 Measures of Hyperﬁnite Energy Integrals

We have defined an inner product ·, · on the hyperfinite dimensional space H by (2.2.3) in Sect. 2.2 or (1.5.15) in Sect. 1.5, Chap. 1.Let||·|| be the norm generated by ·, ·.DenotebyFin(H) the set of all elements in H with finite norm || · ||. By defining u ≈ v if ||u − v|| ≈ 0, we recall that the space

◦H = Fin(H)/ ≈ is a Hilbert space with the inner product (◦u, ◦v)=st(u, v),where◦u denotes the equivalence class of u.

In this section, we shall continue our discussion of Sects. 2.2, 2.3,and2.4. We assume that X and Xˆ are dual hyperﬁnite Markov chains. Let E(·, ·) and Eˆ(·, ·) be the hyperﬁnite quadratic form and co-form associated with X and Xˆ, respectively. 92 2 Potential Theory

We know that for α ∈ ∗R,α≥ 0 and δ ∈ T,

(δ) (δ) Eα (u, v)=E (u, v)+αu, v.

Each of these forms generates a norm (possibly a semi-norm in the case α (δ) (δ) 1 (δ) =0):|u|α =[Eα (u, u)] 2 . Similarly, we denote by Finα (H) the set of all (δ) (δ) (δ) elements in H with ﬁnite norm |·|α . Deﬁne u ≈α v if |u − v|α ≈ 0. The space

◦ (δ) (δ) (δ) Hα = Finα (H)/ ≈α is a Hilbert space if ◦α>0 with respect to the inner product

◦ (δ) (δ) (δ) (δ) ([u]α , [v]α )α = Eα (u, v) ,

(δ) (δ) where [u]α denotes the equivalence class of u under the norm |·|α , and (δ) (·, ·)α denotes the related inner product.

Deﬁnition 2.5.1. Let μ be a hyperﬁnite positive measure on S0.Forδ ∈ T , ∗ ∗ if there exists a constant K ∈ [0, ∞)= R+ such that

N |u(s)| μ(ds)= |u(si)|μ(i) S0 i=1 1 (δ) 2 ≤ K E1 (u, u) (2.5.1) for every u ∈ H, we say that μ is of δ-hyperﬁnite energy integral.Moreover, ∗ ◦ if there exists K ∈ R+ satisfying (2.5.1)and K<∞, μ is said to be of δ-ﬁnite energy integral.

Henceforth, we will identify a hyperﬁnite measure μ on S0 with the measure μ˜ on S deﬁned by μ˜(s0)=0, μ˜(si)=μ(si) for all si ∈ S0.

Theorem 2.5.1. Let E(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form. Then

(i) A positive hyperﬁnite measure μ on S0 is of δ-hyperﬁnite energy integral if and only if for each α ∈ ∗R, 0 < st(α) < ∞, there exists an element (δ) Uα μ ∈ H such that for every v ∈ H, (δ) (δ) Eα (Uα μ, v)= v(s) μ(ds). (2.5.2) S0

(δ) (δ) Moreover, if μ is of δ-ﬁnite energy integral, we have Uα μ ∈ Finα (H). 2.5 Measures of Hyperﬁnite Energy Integrals 93

(ii) A positive hyperﬁnite measure μ on S0 is of δ-hyperﬁnite energy integral if and only if for each α ∈ ∗R, 0 < st(α) < ∞, there exists an element ˆ (δ) Uα μ ∈ H such that for every v ∈ H, (δ) ˆ (δ) Eα (v, Uα μ)= v(s) μ(ds). S0

ˆ (δ) (δ) Moreover, if μ is of δ-ﬁnite energy integral, we have Uα μ ∈ Finα (H).

Proof. The theorem is an easy consequence of Riesz’s representation theorem. (δ) ˆ (δ) Remark 2.5.1. We call Uα μ and Uα μ in Theorem 2.5.1 hyperfinite α- potential and hyperfinite α-co-potential of μ associated with E(δ)(·, ·), respectively. An internal element u ∈ H is called a hyperfinite α-potential (or (δ) ˆ (δ) hyperfinite α-co-potential)ifu = Uα μ (or u = Uα μ) for some positive measure μ of δ-hyperfinite energy integral.

Definition 2.5.2. Fix α ∈ ∗R,α≥ 0 and δ ∈ T. (i) An element u ∈ H is called hyperfinite pre-α-excessive associated with (δ) δ E (·, ·), if u(i) ≥ 0,Q u(i) ≤ (1 + αδ)u(i) for every i ∈ S0 such that m(i) =0. (ii) An element u ∈ H is called hyperfinite pre-α-co-excessive associated with ˆ(δ) ˆδ E (·, ·), if u(i) ≥ 0, Q u(i) ≤ (1 + αδ)u(i) for every i ∈ S0 such that m(i) =0.

(δ) ∗ In order to develop our theory, we shall denote by {Gβ | β ∈ ( −∞, 0)} ˆ(δ) ∗ (δ) and {Gβ | β ∈ ( −∞, 0)} the resolvent and co-resolvent of E (·, ·), respectively. That is, they are deﬁned by

(δ) (δ) −1 Gβ =(A − β) , ˆ(δ) ˆ(δ) −1 Gβ =(A − β) .

∗ Hence, we have for α ∈ R+, δ (δ) (δ) (δ) 1+αδ − Q G−α = δ(α + A )G−α = δ

and ˆδ ˆ(δ) ˆ(δ) ˆ(δ) 1+αδ − Q G−α = δ(α + A )G−α = δ. (2.5.3) 94 2 Potential Theory

Theorem 2.5.2. For δ ∈ T,α ∈ ∗R,α ≥ 0, and u ∈ H, the following conditions are equivalent: (i) u is hyperﬁnite pre-α-excessive associated with E(δ)(·, ·). (ii) There exists a hyperﬁnite positive measure μ on S0 such that (δ) Eα (u, v)= v(s) μ(ds) for all v ∈ H. S0

(δ) (iii) Eα (u, v) ≥ 0 for all v ∈ H, v ≥ 0. If E(δ)(·, ·) is a hyperfinite weak coercive quadratic form, the above statements are equivalent to the following: (iv) u is a hyperfinite α-potential of E(δ)(·, ·). Proof. (i) =⇒ (ii). Assume that u is hyperfinite pre-α-excessive associated (δ) with E (·, ·). Define a hyperfinite positive measure μ on S0 by

μ(s0)=0, 1 δ μ(si)= (1 + αδ)u(i) − Q u(i) m(i) for i ∈ S0. δ Since 1 δ μ(si)= (1 + αδ)u(i) − Q u(i) m(i) δ = A(δ)u(i)+αu(i) m(i),

we see that for every v ∈ H,

N (δ) (δ) Eα (u, v)= A u(i)+αu(i) v(i)m(i) i=1 = v(s) μ(ds). S0

(ii) =⇒ (iii). This is easy, and thus we omit it!

(iii) =⇒ (i).SuchasinSect.1.5 of Chap. 1,letu+ = u ∨0.Sinceu+ − u ≥ 0, (δ) + we have Eα (u, u − u) ≥ 0. Taking into account Corollary 1.5.2, we have

(δ) + + (δ) + + (δ) + Eα (u − u, u − u)=Eα (u ,u − u) −Eα (u, u − u) (δ) + + ≤Eα (u ,u − u) = −E(δ)((−u) ∧ 0, −u − (−u) ∧ 0) ≤ 0. 2.5 Measures of Hyperﬁnite Energy Integrals 95

This implies that

(δ) + + Eα (u − u, u − u)=0.

Therefore, u(i)=u(i) ∨ 0 ≥ 0 for every i ∈ S0 such that m(i) =0.

Furthermore, it follows from (2.5.3) that for any v ∈ H,

(1 + − δ) = (1 + − ˆδ) αδ Q u, v u, αδ Q v (δ) ˆδ ˆ(δ) = Eα u, (1 + αδ − Q )G−αv (δ) = Eα (u, δv). (2.5.4)

Fix i ∈ S0. Let v ∈ H be an internal function deﬁned by v(l)=δil,l ∈ S. Then, we have from (2.5.4)that δ (δ) 1+αδ − Q u(i)m(i)=Eα (u, δv) ≥ 0.

Therefore, the internal function u is hyperﬁnite pre-α-excessive associated with E(δ)(·, ·).

If E(δ)(·, ·) is a hyperﬁnite weak coercive quadratic form, (ii) is equivalent to (iv) by Theorem 2.5.1 (i).

Similarly, we have

Theorem 2.5.3. For δ ∈ T, α ∈ ∗R,α ≥ 0,and u ∈ H, the following conditions are equivalent: (i) u is hyperﬁnite pre-α-co-excessive associated with E(δ)(·, ·). (ii) There exists a hyperﬁnite positive measure μ on S0 such that (δ) Eα (v, u)= v(s) μ(ds) for all v ∈ H. S0

(δ) (iii) Eα (v, u) ≥ 0 for all v ∈ H, v ≥ 0. If E(δ)(·, ·) is a hyperﬁnite weak coercive quadratic form, the above statements are equivalent to the following: (iv) u is a hyperﬁnite α-co-potential of E(δ)(·, ·).

We denote by τ0(δ) the family of all internal positive measures on S0 of δ-hyperﬁnite energy integrals. Denote τ0 = ∪{τ0(δ) | δ is inﬁnitesimal,δ ∈ T }.

Proposition 2.5.1. Let E(δ)(·, ·) be a hyperfinite weak coercive quadratic form. For δ ∈ T, a hyperfinite positive measure μ on S0 is of δ-hyperfinite 96 2 Potential Theory

∗ energy integral if and only if for any α ∈ R+, there exists a hyperﬁnite pre-α-excessive function u associated with E(δ)(·, ·) such that 1 δ μ(i)= (1 + αδ)u(i) − Q u(i) m(i) for all i ∈ S0. (2.5.5) δ

Moreover, if u¯ ∈ H satisﬁes above equation also, then u(i)=¯u(i) for all i ∈ S0 with m(i) =0.

Proof. ⇐= Let u satisfy the condition (2.5.5). Then, we have (δ) Eα (u, v)= v(s) μ(ds) for all v ∈ H. S0

(δ) =⇒ Assume that μ is of δ-hyperﬁnite energy integral. Let u = U1 μ ∈ H satisfy the condition (2.5.2). Then, we have

−1 (δ) μ(i)= G−1 u(i)m(i) 1 = (1 + δ)u(i) − Qδu(i) m(i). δ

∗ Therefore, we have for any α ∈ R+,v ∈ H, −1 (δ) v(s) dμ(s)= v(i) G−1 u(i) dm(i) S S (δ) (δ) (δ) −1 = Eα G−α((G−1) u),v .

(δ) (δ) −1 Hence, w = G−α((G−1) u) is hyperﬁnite pre-α-excessive associated with E(δ)(·, ·) by Theorem 2.5.2.Furthermore,wehave 1 δ μ(i)= (1 + αδ)w(i) − Q w(i) m(i) for all i ∈ S0. δ

Proposition 2.5.2. Let E(δ)(·, ·) be a hyperfinite weak coercive quadratic form. For δ ∈ T, let ν be a hyperfinite positive measure on S0. Define a measure μ on S0 by

μ(s)=ν(s)1(m(s) =0) for s ∈ S0.

Then μ is of δ-hyperﬁnite energy integral. 2.5 Measures of Hyperﬁnite Energy Integrals 97

Proof. Deﬁne

ν(s) f(s)= 1(m(s) =0) for s ∈ S0,f(s0)=0. m(s)

For any u ∈ H,wehave

u(s) dμ(s)= u(s)1(m(s) =0) dν(s) S S = u(s)f(s) dm(s) S (δ) (δ) = E1 (G−1f,u).

Hence, μ is of δ-hyperﬁnite energy integral by Theorem 2.5.2.

We recall that in Sect. 1.4 of Chap. 1, we have introduced the standard part (E(δ)(·, ·),D(E(δ))) on ◦H for a hyperﬁnite weak coercive quadratic form E(δ)(·, ·).

Theorem 2.5.4. Let E(δ)(·, ·) be a hyperfinite weak coercive quadratic form. For α ∈ ∗R, 0 < st(α) < ∞ and δ ∈ T ,letμ be a hyperfinite positive measure of δ-finite energy integral, and let u be an α-potential of μ associated with E(δ)(·, ·). Define (δ) gn(i)=n u(i) − nG−n−αu(i) ,n∈ N.

(δ) Then for every v ∈ Finα (H), we have

◦ (δ) (δ) (δ) ◦ ◦ (i) Eα (G−αgn,v) −→ Eα ( u, v) as n →∞. (2.5.6) (δ) δ (δ) (ii) If u ∈D(E ), let v˜ ≈α v and v˜ ∈D(E ), ◦ ◦ limn→∞ gn(i)v(i) dm(i)= v˜(i) dμ(i). S0 S0

Proof. (i) First of all, we have

◦ (δ) (δ) ◦ Eα (G−αgn,v)= (gn,v) ◦ 2 (δ) = (nu − n G−n−αu, v) ◦ 2 (δ) = (n + α)u − (n + α) G−n−αu, v (2.5.7) ◦ 2 (δ) (δ) + −αu + α G−n−αu +2nαG−n−αu, v .

It follows from Theorem 1.4.1 that 98 2 Potential Theory ◦ 2 (δ) (n + α)u − (n + α) G−n−αu, v + αu, v (δ) ◦ ◦ −→ Eα ( u, v) as n →∞. (2.5.8)

By Lemma 1.4.4,wehave

◦ 2 (δ) α G−n−αu, v −→ 0 as n −→ ∞ . (2.5.9)

Besides, it follows from Theorem 1.4.1 ◦ (δ) −2αu +2nαG−n−αu, v ◦ 2 ( + )2 ≤ αn − n α +( + ) ( + )2 u n α u, v n α n 2 (δ) +−(n + α)u +(n + α) G−n−αu, v −→ 0 as n →∞. (2.5.10)

From the relations (2.5.7), (2.5.8), (2.5.9), and (2.5.10), we know that the approximation (2.5.6)holds.

(ii) From the proof of (i), we get

◦ ◦ (δ) lim gn(s)v(s) m(ds)= E (u, v˜) n→∞ α S0 ◦ = v˜(s) μ(ds). S0 Proposition 2.5.3. Let E(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C.Forδ ∈ T ,letμ be a positive measure of δ-hyperﬁnite energy integral. Then for every L(μ) measurable subset A of S0, we have

◦' ( ) ( (δ) (δ) (δ) ◦ (δ) L(μ)(A) ≤ C E1 (U1 μ, U1 μ) Cap1 (A). (2.5.11)

◦ (δ) Proof. For simplicity, we assume that Cap1 (A) < ∞. If A is internal, we have

◦ ◦ μ(A)= 1A(s) μ(ds) S0 ◦ (δ) ≤ e1 (A)(s) μ(ds) S0 2.5 Measures of Hyperﬁnite Energy Integrals 99

◦ (δ) (δ) (δ) = E1 (U1 μ, e1 (A)) ◦' ( ( ) (δ) (δ) (δ) (δ) (δ) (δ) ≤ C E1 (U1 μ, U1 μ) E1 (e1 (A),e1 (A))

◦' ( ( ) (δ) (δ) (δ) (δ) = C E1 (U1 μ, U1 μ) Cap1 (A) .

Now it is easy to see that the inequality (2.5.11)holdsforallL(μ) measurable subsets A.

Corollary 2.5.1. Let E(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form. Let μ be a positive measure of δ-ﬁnite energy integral on S0.ThenL(μ) charges no set of δ-zero capacity. Proof. The proof follows easily from Proposition 2.5.3.

We shall now state and prove the following characterization theorem.

Theorem 2.5.5. Let A ⊂ S0 be an A(S0)-measurable set (in particular, A ∈ σ(S0)) and let δ ∈ T. We have

◦ (δ) (i) For any μ ∈ τ0(δ),L(μ)(A)=0=⇒ Cap1 (A)=0. ◦ (δ) (ii) For any μ ∈ τ00(δ),L(μ)(A)=0=⇒ Cap1 (A)=0, where ◦ (δ) τ00(δ)= μ ∈ τ0(δ)μ(S0)=1, ||U1 μ||∞ < ∞

and (δ) (δ) ||U1 μ||∞ =max |U1 μ(s)|s ∈ S0 .

◦ (δ) (iii) For any μ ∈ τˆ00(δ),L(μ)(A)=0=⇒ Cap1 (A)=0, where ◦ ˆ (δ) τˆ00(δ)= μ ∈ τ0(δ)μ(S0)=1, ||U1 μ||∞ < ∞

and ˆ (δ) ˆ (δ) U1 μ ∞=max |U1 μ(s)|s ∈ S0 .

◦ (δ) Proof. (i) Assume that ∞≥ Cap1 (A)=α>0. Since A ∈A(S0), it follows from Theorem 2.3.1 (ii) that A is capacitable with respect to the capacity ◦ (δ) Cap1 (·). That is ◦ (δ) ◦ (δ) Cap1 (A)=sup Cap1 (B)B = Bn,Bn ∈S0 and B ⊂ A . n∈N 100 2 Potential Theory

Therefore, there exists a sequence {Bn | n ∈ N} of decreasing internal subsets of S0 such that ∞ ∞ ⊂ ∞≥◦Cap(δ) ≥ α 0 Bn A, 1 Bn 2 > . n=1 n=1

∗ Let {Bn | n ∈ N} be an internal decreasing extension of {Bn | n ∈ N}. Then, there exists an inﬁnite γ ∈ ∗N − N such that ∞ ⊂ 0 α ≤ ◦Cap(δ) = ◦Cap(δ)( ) ≤∞ Bγ A, < 2 1 Bn 1 Bγ . n=1

(δ) Set B = Bγ . Consider the internal function e1 (B). Notice that if i/∈ B, then (δ) (δ) −σB /δ (1 + δ)e1 (B)(i)=(1+δ)Ei (1 + δ) N (δ) (δ) = e1 (B)(j)qij j=1 δ (δ) = Q e1 (B)(i), (2.5.12)

(δ) where σB =min{t ∈ Tδ | X(ω,t) ∈ B}. If i ∈ B, then we have

(δ) (1 + δ)e1 (B)(i)=(1+δ) N (δ) (δ) ≥ e1 (B)(j)qij j=1 δ (δ) = Q e1 (B)(i). (2.5.13)

It follows from the relations (2.5.12)and(2.5.13) that the function (δ) (δ) e1 (B) is hyperfinite 1-excessive associated with E (·, ·). Define a hyperfinite positive measure μ on S by

μ(s0)=0, 1 (δ) δ (δ) μ(si)= (1 + δ)e (B)(i) − Q e (B)(i) m(i),i∈ S0. δ 1 1 Then, we have μ(B)= μ(ds) S0 (δ) (δ) = E1 (e1 (B), 1) (δ) =Cap1 (B). 2.5 Measures of Hyperﬁnite Energy Integrals 101

This contradicts the assumption ◦L(μ)(A)=0. Hence, we must have ◦ (δ) Cap1 (A)=0.

◦ (δ) (ii) Assume that 0 < Cap1 (A)=α ≤∞. By the proof (i), we see that there exists an element ν ∈ τ0(δ) such that (replacing μ in the proof of (i) by ν)

◦ (δ) (δ) 0 < Cap1 (B) ≤∞, Cap1 (B)=ν(B)=ν(S0) (δ) (δ) and U1 ν(i)=e1 (B)(i) ≤ 1

for any i ∈ S0,whereB is an internal set contained in A. Deﬁne μ(·) by

(·) (·)= ν μ (δ) . Cap1 (B)

Then, we have

μ(S0)=1,μ∈ τ0(δ) and ◦ ◦ | (δ) ( )| max | (δ) ( )| = max U1 ν i U1 μ i (δ) i∈S0 i∈S0 Cap (B) 1 ◦ 1 ≤ (δ) Cap1 (B) < ∞.

But

1=◦μ(B) ≤ ◦μ(A) ≤ 1, i.e., L(μ)(A)=1.

This contradicts our assumption of L(μ)(A)=0. We have proved that

◦ (δ) Cap1 (A)=0.

(iii) Recalling Corollary 2.2.1, we can show this result in the same way as that of above (ii).

Theorem 2.5.6. For δ ∈ T ,letE(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form, which has the following property:

◦ (δ) ∀A ⊂ S0, Cap1 (A)=∞ ◦ (δ) =⇒∃B ⊂ A such that 0 < Cap1 (B) < ∞. (2.5.14) 102 2 Potential Theory

Let τ0f (δ) be the family of all positive measures of δ-ﬁnite energy integrals. Then the following statements are equivalent for A ∈A(S0) (in particular, A ∈ σ(S0)): ◦ (δ) (i) A is of zero δ-capacity, i.e., Cap1 (A)=0. (ii) For any μ ∈ τ0f (δ),L(μ)(A)=0. (iii) For any μ ∈ τ00f (δ),L(μ)(A)=0, where ◦ (δ) τ00f (δ)= μ ∈ τ0f (δ)μ(S0)=1, ||U1 μ||∞ < ∞ .

(iv) For any μ ∈ τˆ00f (δ),L(μ)(A)=0, where ◦ ˆ (δ) τˆ00f (δ)= μ ∈ τ0f (δ)μ(S0)=1, ||U1 μ||∞ < ∞ .

Proof. (i)=⇒ (ii)=⇒ (iii) and (ii)=⇒ (iv) are clear by Corollary 2.5.1.We can show (ii)=⇒ (i) and (iii)=⇒ (i) and (iv)=⇒ (i) in the same way as the proof of Theorem 2.5.5.

2.6 Internal Additive Functionals and Associated Measures

Let X and Xˆ be dual hyperfinite Markov chains, and let E(·, ·) and Eˆ(·, ·) be the hyperfinite quadratic form and co-form of X and Xˆ, respectively. Again, let H be the hyperfinite dimensional space with an inner product ·, · defined by (2.2.3) in Sect. 2.2 or (1.5.15) in Sect. 1.5, Chap. 1.

We recall that in Sect. 1.5 of Chap. 1, we have introduced the dual (δ) (δ) hyperfinite Markov chains (Ω,X , {Ft | t ∈ Tδ}, {Pi | i ∈ S}) and ˆ ˆ (δ) ˆ(δ) ˆ (Ω,X , {Ft | t ∈ Tδ}, {Pi | i ∈ S}) for δ ∈ T . For simplicity, we assume ˆ (δ) ˆ (δ) (δ) ˆ(δ) that Ω = Ω, X = X , Ft = Ft ,t ∈ Tδ. The hyperfinite quadratic ˆ ˆ form associated with (X, {Pi | i ∈ S}) (or (X,{Pi | i ∈ S})) are given by the expression (1.5.19)(or(1.5.22)) in Sect. 1.5. As in the study of standard Markov processes, we define a family of translation operators {θt | t ∈ T } of Ω.Thatis,foreacht ∈ T,θt is a map from Ω to Ω defined by

ω ∈ Ω =⇒ θtω ∈ Ω and for any s ∈ T,θtω(s)=ω(s + t).

(δ) Hence for each δ ∈ T , we have a family of translation operators {θt | t ∈ Tδ} (δ) induced by {θt | t ∈ T },θt = θt for any t ∈ Tδ. In other words, for each (δ) t ∈ Tδ,θt is a map from Ω to Ω given by

(δ) (δ) ω ∈ Ω =⇒ θt ω ∈ Ω and for any s ∈ Tδ,θt ω(s)=ω(s + t). 2.6 Internal Additive Functionals and Associated Measures 103

Deﬁnition 2.6.1. For any δ ∈ T, we call an internal ∗R-valued function A(ω,t) or At(ω),t ∈ Tδ,ω ∈ Ω, δ-internal additive functional (abbreviated by δ-IAF) if it satisﬁes the following two conditions:

(1) For each t ∈ Tδ,At(ω) is non-anticipating with respect to the ﬁltration (δ) (δ) (Ω,{Ft t ∈ Tδ}), i.e., At(·) is Ft -measurable, ∀t ∈ Tδ. (2) For each ω ∈ Ω, we have

A(ω,0) = 0, (δ) A(ω,t + s)=A(ω,s)+A(θs ω,t) for any t, s ∈ Tδ.

Proposition 2.6.1. If A(ω,t) is a δ-internal additive functional, then there exists a hyperﬁnite measure μ on S0 (not necessarily positive) such that ∗ μ (i)=0whenever m(i)=0and for all n ∈ N,f, h ∈ H, n (δ) h(i)Ei f(X (kδ)) A(ω,(k +1)δ) − A(ω,kδ) dm(i) S0 k=0 n ˆ (δ) = f(i)Eih(X (kδ)) dμ(i)δ. (2.6.1) k=0 S0

Proof. Deﬁne

μ (0) = 0, 1 μ (i)= EiA(ω,δ)m(i), 1 ≤ i ≤ N. (2.6.2) δ Then, we have n (δ) h(i)Ei f(X (kδ)) A(ω,(k +1)δ) − A(ω,kδ) dm(i) S0 k=0 n (δ) (δ) = h(i)Eif(X (kδ, ω))A(θkδ ω,δ) dm(i) k=0 S0 n = h(i)Eif(X(kδ))EX(kδ)A(ω,δ) dm(i) k=0 S0 n ˆ = f(i)EiA(ω,δ)Eih(X(kδ)) dm(i) k=0 S0 n ˆ = f(i)Eih(X(kδ)) dμ(i)δ. k=0 S0 104 2 Potential Theory

Proposition 2.6.2. Let μ be a hyperﬁnite measure on S0 satisfying μ(i)=0 whenever m(i)=0. Then for each δ ∈ T ,thereexistsaδ-IAF A(ω,t) such that (2.6.1) holds.

( )= μ(s) Proof. First of all, let f s m(s) 1(m(s) =0). For each u ∈ H,wehave u(s) dμ(s)= u(s)f(s) dm(s) S0 S0 (δ) ˆ(δ) = E1 (u, G−1f).

Deﬁne

A(ω,0) = 0, k A(ω,kδ)=δ f(X(δ)(ω,(l − 1)δ)) for k ∈ ∗N,k ≥ 1. l=1

It is easy to verify that A(ω,t) is a δ-IAF. Moreover, we have for each i ∈ S0,

1 1 (δ) EiA(ω,δ)m(i)= Eif(X (ω,0))δm(i) δ δ = f(i)m(i) = μ(i).

Therefore, it follows from the proof of Proposition 2.6.1 that (2.6.1)holds.

For δ ∈ T ,letA(ω,t) be a δ-IAF. Deﬁne 1 e(A)= E(A(ω,δ))2 2δ 1 = 2 ( ) 2 EiAδ dm i . (2.6.3) δ S0

We call e(A) the energy of A. Furthermore, we deﬁne the mutual energy e(A, B) for δ-internal additive functionals A and B by 1 e(A, B)= E A(ω,δ)B(ω,δ) . 2δ

Let ΔA(ω,kδ) be the forward increment of A(ω,t) at time kδ, i.e.,

ΔA(ω,kδ)=A(ω,(k +1)δ) − A(ω,kδ) for k ∈ ∗N.

∗ We deﬁne the quadratic variation [A]:Ω × Tδ −→ R by 2.6 Internal Additive Functionals and Associated Measures 105

[A](ω,0) = 0, n−1 2 [A](ω,nδ)= (ΔA(ω,kδ)) for n ∈ ∗N,n>0. k=0

Because n−1 n+m−1 2 2 [A](ω,(n + m)δ)= (ΔA(ω,kδ)) + (ΔA(ω,kδ)) k=0 k=n n−1 m−1 2 2 = (ΔA(ω,kδ)) + (ΔA(ω,(k + n)δ)) k=0 k=0 n−1 m−1 2 2 = (ΔA(ω,kδ)) + (ΔA(θnδω,kδ)) , k=0 k=0

[A] is a positive δ-IAF. By Proposition 2.6.1 and its proof, we know that 1 2 μ<[A]>(i)= δ Ei(A(ω,δ)) m(i) is the hyperﬁnite positive measure associated with [A] in the sense (2.6.1). We call μ<[A]> the energy measure of A. It is obviously from the relations (2.6.2)and(2.6.3)that 1 ( )= ( ) e A 2μ<[A]> S0 . (2.6.4)

Let u ∈ H.Forδ ∈ T , deﬁne a δ-IAF A[u](ω,t) by

[u] (δ) (δ) A (ω,t)=u(X (ω,t)) − u(X (ω,0)) for t ∈ Tδ.

Then, we have

1 2 e(A[u])= E A[u](ω,δ) 2δ 1 2 = E u(X(δ)(ω,δ)) − u(X(δ)(ω,0)) 2δ 1 N 2 = ( (δ)( )) − ( ) ( ) 2 Ei u X ω,δ u i m i δ i=0 1 N N = ( ( ) − ( ))2 (δ) ( ) 2 u j u i qij m i δ i=0 j=0 1 N N 1 N = ( ( ) − ( ))2 (δ) ( )+ ( ( ))2 (δ) ( ) 2 u j u i qij m i 2 u i qi0 m i δ i=1 j=1 δ i=1 106 2 Potential Theory

1 N N 1 N = ( ( ) − ( )) ( ) (δ) ( )+ ( ( ))2 (δ) ( ) 2 u i u j u i qij m i 2 u i qi0 m i δ i=1 j=1 δ i=1 1 N N 1 N + ( ( ) − ( )) ( )ˆ(δ) ( )+ ( ( ))2 ˆ(δ) ( ) 2 u j u i u j qji m j 2 u i qi0 m i δ i=1 j=1 δ i=1 1 N − ( ( ))2 ˆ(δ) ( ) 2 u i qi0 m i δ i=1 1 N = E(δ)( ) − ( ( ))2 ˆ(δ) ( ) u, u 2 u i qi0 m i , (2.6.5) δ i=1 wherewehaveusedLemma1.5.1 (i) and (ii) in the last equation of (2.6.5).

Theorem 2.6.1. For u ∈ H, f ∈ H, let μ(·) be the energy measure of A[u]. We have

2 1 [u] (1) μ(i)= Ei A (ω,δ) m(i) δ N 1 2 (δ) = (u(j) − u(i)) q m(i). δ ij j=0 (δ) (δ) 2 (2) f(s) μ(ds)=2E (u, uf) −E (u ,f). (2.6.6) S0

Proof. (1) The proof is quite immediate, and hence we omit it!

(2) On the one hand, we have N N 2 1 (δ) f(s) μ(ds)= f(i) u(j) − u(i) qij m(i). (2.6.7) S0 δ i=1 j=0

On the other hand, we have

2E(δ)( ) −E(δ)( 2 ) u, fu⎡ u ,f ⎤ 2 N N ⎣ 2 (δ) ⎦ = (u(i)) f(i)m(i) − (uf)(i)u(j)qij m(i) δ i=1 j=1 ⎡ ⎤ 1 N N ⎣ 2 2 (δ) ⎦ − (u(i)) f(i)m(i) − (u(j)) f(i)qij m(i) δ i=1 j=1 N N N 1 2 1 (δ) = (u(i)) f(i)m(i) − (2u(i) − u(j))u(j)f(i)qij m(i) δ i=1 δ i=1 j=1 2.7 Fukushima’s Decomposition Theorem 107 ⎛ ⎞ 1 N N ⎝ 2 (δ)⎠ = f(i)m(i) (u(i)) − (2u(i) − u(j))u(j)qij δ i=1 j=0 N N 1 2 (δ) = f(i)(u(j) − u(i)) qij m(i). (2.6.8) δ i=1 j=0

By (2.6.7)and(2.6.8), we get (2.6.6).

2.7 Fukushima’s Decomposition Theorem

As before, we shall work under the conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14) of Sect. 1.5, Chap. 1, in this section.

2.7.1 Decomposition Under the Individual Probability Measures Pi

Lemma 2.7.1. For δ ∈ T ,letE(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C.Letν be a positive measure on S0 of δ-hyperﬁnite energy integral. For any u ∈ H, t ∈ Tδ,andε>0, we have t (δ) Pν ω∃s ∈ Tδ (|u(X (ω,s))|≥ε)

2 t/δ 1 2C (1 + δ) (δ) (δ) (δ) (δ) 2 ≤ E (U ν, U ν)E (u, u) , ε 1 1 1 1 where Pν (·)= Pi(·)dν(i). S0

Proof. Let A = {i ∈ S0 | u(i) ≥ ε} . Deﬁne

(δ) (δ) σA (ω)=min{t ∈ Tδ | X (ω,t) ∈ A}.

Then, we have (δ) Pν ω∃s ∈ Tδ,s≤ t u(X (ω,s)) ≥ ε (δ) −σA /δ −t/δ = Pi ω (1 + δ) ≥ (1 + δ) dν(i) S 0 −σ(δ) /δ t/δ ≤ Ei (1 + δ) A (1 + δ) dν(i) S0 108 2 Potential Theory t/δ (δ) =(1+δ) e1 (A)(i) dν(i) S0 t/δ (δ) (δ) (δ) =(1+δ) E1 U1 ν, e1 (A)

1 t/δ (δ) (δ) (δ) (δ) (δ) (δ) 2 ≤ C (1 + δ) E1 U1 ν, U1 ν E1 e1 (A),e1 (A)

2 t/δ 1 C (1 + δ) (δ) (δ) (δ) (δ) 2 ≤ E U ν, U ν E (u, u) , ε 1 1 1 1 where the reason for the last step holding is (δ) (δ) (δ) (δ) | u | | u | E e (A),e (A) ≤ C2E , 1 1 1 1 ε ε 2 C (δ) ≤ E (u, u), ε2 1 by Corollary 2.2.3. Hence, we can prove Lemma 2.7.1 by applying the same argument to −u.

Proposition 2.7.1. For δ ∈ T, let E(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Assume that (E(δ)(·, ·), D(E(δ))) satisﬁes condition (2.5.14) of Theorem 2.5.6.Letu, {un | n ∈ N} be the elements in H and let δ ∈ T. Suppose that

◦ (δ) E1 (un − u, un − u) −→ 0 as n →∞.

{ | ∈ N} Then there exist a subsequence unk k and a δ-exceptional set B such ﬁn that for all i ∈ S0 − B,t ∈ Tδ , ( ) ◦ ( (δ)( )) ◦ ( (δ)( )) L Pi unk X ω,s converges uniformly to u X ω,s t in s on Tδ as n −→ ∞ =1.

Proof. Let {nk | k ∈ N} be a subsequence satisfying

◦E(δ)( − − ) ≤ 2−4k 1 unk u, unk u .

Set ( )= ∃ ∈ t | ( (δ)( )) − ( (δ)( ))|≥2−k Λk t ω s Tδ unk X ω,s u X ω,s .

For ν ∈ τ00f (δ), we have from Lemma 2.7.1 that 2.7 Fukushima’s Decomposition Theorem 109 ◦ 1 ◦ 2 t −k+1 (δ) (δ) (δ) 2 Pν (Λk(t)) ≤ C (1 + δ) δ 2 E1 (U1 ν, U1 ν) .

Hence, we have

∞ L(Pν )(Λk(t)) < ∞. k=1

By the Borel-Cantelli lemma, we get ∞ ∞ L(Pν ) Λl(t) =0. (2.7.1) k=1 l=k

∞ ∞ Set Λ(t)= k=1 l=k Λl(t). From Theorem 2.5.6,Theorem2.4.1,and(2.7.1), there exists a δ-exceptional set B(t) such that

L(Pi)(Λ(t)) = 0 for any i ∈ S0 − B(t).

Now let us select a countable subset {tn | n ∈ N}⊂Tδ such that tn ≈ n. We ∞ ∞ deﬁne Λ = n=1 Λ(tn),B = n=1 B(tn). It is easy to see that Proposition 2.7.1 holds.

Lemma 2.7.2. For δ ∈ T, let A be a δ-IAF and let μ(i) be the hyperﬁnite measure deﬁned by (2.6.2)inSect.2.6. Then for any v ∈ H, we have (δ) ˆ (δ) E (ft,v)= v(i) − Eiv(X (t)) dμ (i) for all t ∈ Tδ, S0 (2.7.2) where ft(i)=EiA(ω,t).

Proof. If k =1,thenwehave (δ) 1 ˆδ E (fδ,v)= v(i) − Q v(i) fδ(i) dm(i) δ S 0 ˆ (δ) = v(i) − Eiv(X (δ)) dμ (i). S0

Assume that (2.7.2) holds whenever k ≤ n. Then, we have (δ) 1 ˆδ E (f(n+1)δ,v)= v(i) − Q v(i) EiA(ω,(n +1)δ) dm(i) δ S 0 1 ˆδ (δ) = v(i) − Q v(i) EiA(ω,nδ)+EiA(θnδ ω,δ) dm(i) δ S0 110 2 Potential Theory ˆ (δ) = v(i) − Eiv(X (nδ)) dμ (i) S 0 1 ˆ (δ) ˆ (δ) + Eiv(X (nδ)) − Eiv(X ((n +1)δ)) δ S0 ×EiA(ω,δ) dm(i) ˆ (δ) = v(i) − Eiv(X ((n +1)δ)) dμ (i). S0

Lemma 2.7.3. For δ ∈ T ,letE(δ)(·, ·) be a hyperfinite weak coercive quadratic form with continuity constant C.LetA(ω,t) be a positive δ-IAF. For all positive hyperfinite measures ν on S0 of δ-hyperfinite energy integral, we have

ˆ (δ) Eν (At) ≤ (1 + t)||U1 ν||∞μ (S0) for all t ∈ Tδ. (2.7.3)

Proof. It follows from Theorem 2.5.1 and Lemma 2.7.2 that (δ) ˆ (δ) Eν A(ω,t)= EiA(ω,t) dν(i)=E1 (ft, U1 ν) S 0 ˆ (δ) ˆ ˆ (δ) (δ) = U1 ν(i) − Ei(U1 ν(X (t))) dμ (i) S0 ˆ (δ) + ft(i)U1 ν(i) dm(i) S0 ' ) ˆ (δ) ≤||U1 ν||∞ μ (S0)+ ft(i) dm(i) , (2.7.4) S0 wherewehaveusedTheorem2.5.3 in the latter inequality above. Notice that

f(k+1)δ(i) dm(i)= EiA(ω,(k +1)δ) dm(i) S0 S0

= EiA(ω,kδ) dm(i)+ EiA(θkδω,δ) dm(i) S0 S0 = ( ) ( )+ ˆ 1 EiA ω,kδ dm i Ei (X(kδ)∈S0) S0 S0 × ( ) ( ) EiA δ dm i

≤ EiA(ω,kδ) dm(i)+ EiA(ω,δ) dm(i). S0 S0 2.7 Fukushima’s Decomposition Theorem 111

We can show

ft(i) dm(i) ≤ tμ (S0). (2.7.5) S0

From the relations (2.7.4)and(2.7.5), we obtain the inequality (2.7.3). ∗ Deﬁnition 2.7.1. We call an internal process A : Ω ×Tδ → R a martingale (δ) (δ) with respect to (Ω,Ft ,Pi,i∈ S0) if ω → A(ω,t) is Ft measurable for all (δ) t ∈ Tδ,andforalls, t ∈ Tδ,s

Ei (1B(At − As)) = 0.

(δ) It is easy to see that if [ω]t is the equivalence class of ω deﬁned by (1.5.35) in Sect. 1.5 of Chap. 1, then a non-anticipating process A(ω,t) is a martingale if and only if ΔA(˜ω,t)Pi {ω˜} =0. (2.7.6) (δ) ω˜∈[ω]t

In the following, we shall use “δ-quasi-everywhere” or “δ-q.e.” to mean “except for a δ-exceptional set”. A statement depending on i ∈ S0 is said to be “δ-quasi-everywhere” or “δ-q.e.” if there exists a set B ⊂ S0 of δ-exceptional such that the statement is true for every i ∈ S0 − B.

Let us introduce the following set of internal martingale additive functionals (abbreviated by δ-IMAF) by (δ) ◦ 2 M = M | M is a δ-IAF, (EiM (t)) < ∞,δ-q.e.i ∈ S0 for all ﬁnite t ∈ Tδ and EiM(t)=0, ∀t ∈ Tδ, ∀i ∈ S0} .

(δ) For M ∈M , we have for all s, t ∈ Tδ

(δ) (δ) (δ) Ei(Ms+t |Ft )=Ei(Mt + Ms(θt ) |Ft )

= Mt + EX(t)Ms = Mt,

(δ) for all i ∈ S0. Hence, M ∈M is a martingale with respect to Pi for all (δ) i ∈ S0. Meantime, M ∈M is also square integrable with respect to Pi for δ-q.e. i ∈ S0. In the following, let

M˚ (δ) = {M ∈M(δ) | ◦e(M) < ∞}.

Any element in M˚(δ) is called a δ-internal martingale additive functional of ﬁnite energy. 112 2 Potential Theory

Proposition 2.7.2. For δ ∈ T ,letE(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C.Let{An | n ∈ N} be δ-internal ◦ additive functionals. Assume that e(An − Am) → 0 as n, m →∞, and for (δ) each i ∈ S0, {Ω,Ft ,An(t),Pi} is a martingale for all n ≥ 1. Then there ˚(δ) ◦ ◦ exists a unique A ∈ M with inner product e(·, ·) such that e(An − A) → 0 →∞ { ( ) | ∈ N} as n . Moreover, there exist a subsequence Ank ω,t k and a ﬁn δ-exceptional set B such that for all i ∈ S0 − B,t ∈ Tδ , ( ) ◦ ( ) −→ ◦ ( ) t =1 L Pi ω Ank ω,s A ω,s uniformly on Tδ .

Proof. By the equality (2.6.4) in Sect. 2.6, we know that

◦ ◦ ( )= 2 ( − ) μ<[An−Am]> S0 e An Am −→ 0 as n, m →∞.

◦ Hence, there exists a subsequence {nk | k ∈ N} such that μ<[A −A ]>(S0) nk+1 nk −3k < 2 .SinceAn,n ≥ 1 are martingales with respect to Pi,i ∈ S0,weget ﬁn from Lemma 2.7.3 that for all ν ∈ τˆ00(δ),t∈ Tδ :

◦ ◦ ( ) − ( ) 2 = ([ − ]( )) Eν Ank+1 t Ank t Eν Ank+1 Ank t ◦ ˆ (δ) −3k ≤ (1 + t)||U1 ν||∞2 . (2.7.7)

From the relation (2.7.7) and Doob’s inequality (for which we refer to [25] 4.2.8), we get

◦' ) −k Pν max |An (s) − An (s)|≥2 s≤t k+1 k ◦ 2 2k ≤ 2 Eν max |An (s) − An (s)| s≤t k+1 k ◦ ≤ 22k+2 ( ) − ( ) 2 Eν Ank+1 t Ank t ◦ ˆ (δ) −k ≤ 4(1 + t)||U1 ν||∞2 . (2.7.8)

It is easy to see from the inequality (2.7.8), by using Borel-Cantelli lemma, that

L(Pν )(Λ)=1, for all ν ∈ τˆ00, 2.7 Fukushima’s Decomposition Theorem 113

= { | ◦ ( ·) t} where Λ ω Ank+1 ω, converges uniformly on each finite interval Tδ . By Theorem 2.5.5 and Theorem 2.4.1,wehaveL(Pi)(Λ)=1, δ-q.e.i ∈ S0. Let { | ∈ ∗N} { | ∈ N} ( )= Ank k be an internal extension of Ank k . Define A ω,t ∗ AnK (ω,t) for some K ∈ N − N. Then, A(ω,t) is a Pi-martingale δ-IAF for 2 ◦ 2 all i ∈ S0.SinceAn(t) converges in L (Pν ), we have ( Eν A (t)) < ∞, ∀t ∈ fin (δ) Tδ . Therefore, we have shown A ∈M . For ε>0,chooseN such that e(An −Am) <ε,n,m>N.By Fatou’s lemma, we have e(An −A) ≤ ε, n > N. ˚ (δ) ◦ Thus, we have A ∈ M and e(An − A) −→ 0 as n −→ ∞ . Definition 2.7.2. (1) We call an internal function u ∈ H S-bounded if there is a positive constant C ∈ [0, ∞) such that |u(i)|≤C for all i ∈ S0. ∗ (2) An internal function f : Tδ −→ R is called S-continuous if f(s) ≈ f(t) whenever s ≈ t and s and t are nearstandard. Lemma 2.7.4. For δ ∈ T,u ∈ Fin(H), define a δ-IAF Aδ by

Aδ(ω,0) = 0, n Aδ(ω,nδ)=δ u(X(ω,(k − 1)δ)),n∈ N,n≥ 1. k=1

Then, there exists a properly δ-exceptional set B ⊂ S0 such that for all i ∈ S0 − B, δ L(Pi) ωA (ω,·) is S-continuous =1.

Proof. First of all, we assume that u is S-bounded. We have for any ω ∈ Ω, |Aδ(ω,t) − Aδ(ω,s)|≤|t − s| max |u(i)| +1 . i∈S0

This implies that Aδ(ω,t) is S-continuous. Next, we suppose that u ∈ Fin(H). For each n,set

Bn = {s ∈ S0 ||u(s)|≥n} .

In addition, we deﬁne

σBn (ω)=min{t ∈ Tδ | X(t) ∈ Bn} .

Then, we have 1 P ω∃t ∈ Tδ (X(ω,t) ∈ Bn) = ( ) ≤ 1 P ωσBn ω 114 2 Potential Theory −2 2 ≤ n Ei[u(X(σBn ))] dm(i) S0 ≤ n−2 [u(i)]2 dm(i). S0

Thus, we get ∞ 1 L(P ) ω∃t ∈ Tδ (X(ω,t) ∈ Bn) =0. n=1

◦ This implies that {s ∈ S0 | |u(s)| = ∞} is a δ-exceptional set. Let B be ◦ a properly δ-exceptional set containing {s ∈ S0 | |u(s)| = ∞} (Proposition 2.1.1). For each n ∈ N, deﬁne

σn(ω)=min{t ∈ Tδ ||u(X(ω,t))|≥n} .

δ Then for each ω ∈ Ω,A (ω,·) is continuous in [0,σn(ω)) for every n. Moreover, for each i ∈ S0 − B, we have

◦ L(Pi) {ω | σn(ω) ↑∞ as n →∞}=1, which proves Lemma 2.7.4. Set

(δ) NC = {N | L(Pi){ω | N(ω,·) is an S-continuous δ-IAF} =1, q.e.i ∈ S0 and e(N) ≈ 0} .

(δ) For N ∈NC , the variation vanishes in the following sense

ﬁn E([N](t)) ≈ 0 for all t ∈ Tδ .

In fact, we have for all ﬁnite t ∈ Tδ, E([N](t)) = E(ΔN(ω,s))2 0

Theorem 2.7.1. For infinitesimal δ ∈ T, let E(δ)(·, ·) be a hyperfinite weak coercive quadratic form with continuity constant C. Assume that (E(δ)(·, ·), D(E(δ))) satisfies the property (2.5.14)inSect.2.5. For any u ∈D(E(δ)), there are two δ-internal additive functionals M [u](ω,t) ∈ M˚ (δ) and N [u](ω,t) ∈ (δ) NC such that

[u] [u] [u] A (ω,t)=M (ω,t)+N (ω,t), ∀ω ∈ Ω,∀t ∈ Tδ.

Proof. Step 1. Deﬁne

N [u](ω,0) = 0, ΔN [u](ω,t)=Qδu(X(δ)(ω,t)) − u(X(δ)(ω,t)) (δ) (δ) = −δA u(X (ω,t)),t∈ Tδ, and

M [u](ω,t)=u(X(δ)(ω,t)) − u(X(δ)(ω,0)) − N [u](ω,t).

For each t ∈ Tδ, we have

M [u](ω,t + δ) − M [u](ω,t)=u(X(δ)(ω,t + δ)) − Qδu(X(δ)(ω,t)). (2.7.9)

(δ) [u] It is easy to see from (2.7.6)and(2.7.9)that(Ω,Ft ,M (ω,t),Pi) is a martingale for each i ∈ S0. Furthermore, it is easy to see that

A[u](ω,t)=M [u](ω,t)+N [u](ω,t).

Step 2. Assume that ◦A(δ)u, A(δ)u < ∞. We have from Lemma 2.7.4 that there exists a properly δ-exceptional set B ⊂ S0 such that for all i ∈ S0 − B, [u] L(Pi) ωN (ω,·) is S-continuous =1.

Moreover, we have

2 2 E ΔN [u](t) = δ2E A(δ)u(X(δ)(ω,t) ≤ δ2Au, Au. (2.7.10)

From the relation (2.7.10), we obtain

[u] ﬁn E[N ](t) ≤ tδAu, Au≈0 for each t ∈ Tδ , 116 2 Potential Theory and 1 2 e(N [u])= E N [u](ω,δ) 2δ 1 ≤ 2δ Au, Au ≈ 0. (2.7.11)

Therefore, we deduce from (2.6.5) in Sect. 2.6 and the relation (2.7.11)that

e(M [u])=e(A[u] − N [u]) ≈ e(A[u]) ≤E(δ)(u, u). (2.7.12)

By using Lemma 2.7.3,(2.6.4) in Sect. 2.6,andtherelation(2.7.12), we get ﬁn for any ν ∈ τˆ00(δ),t∈ Tδ ◦ [u] 2 ◦ [u] Eν (M (ω,t)) = Eν [M ](t) ◦ ˆ (δ) ≤ (1 + t)||U1 ν||∞μ<[M [u]]>(S0) ◦ ˆ (δ) [u] = (1 + t)||U1 ν||∞2e(M ) ◦ ˆ (δ) (δ) ≤ (1 + t)||U1 ν||∞2E (u, u) < ∞. (2.7.13)

Therefore, it follows from Theorem 2.5.5 and the relation (2.7.13) that there exists a δ-zero capacity set A1 ⊂ S0 such that ' ) ◦ 2 [u] ﬁn Ei M (ω,t) < ∞ for all t ∈ Tδ ,i∈ S0 − A1.

Now it is easy to see that all results of Theorem 2.7.1 hold whenever

◦A(δ)u, A(δ)u < ∞.

(δ) Step 3. For general u ∈D(E ), put un = nG−nu, n ∈ N. Then

[un] δ ΔN (ω,t)=Q un(X(ω,t)) − un(X(ω,t)) (δ) = −δA un(X(ω,t)). 2.7 Fukushima’s Decomposition Theorem 117

First of all, we have ◦ 2 ◦ 2 (δ) (δ) A un(i) dm(i)= (A + n)un(i) − nun(i) dm(i) S0 S0 ◦ 2 = [nu(i) − nun(i)] dm(i) S0 ◦' ) ≤ 4n2 (u(i))2 dm(i) S0 < ∞. (2.7.14)

Therefore, we know from the relation (2.7.14) and Lemma 2.7.4 that, for each n ∈ N, there exists a properly δ-exceptional set Bn ⊂ S0 such that for all i ∈ S0 − Bn, [un] L(Pi) ωN (ω,·) is S-continuous =1. (2.7.15)

It follows from the relation (2.7.12)that ◦ [um] [un] ◦ (δ) e M − M ≤ E (um − un,um − un) −→ 0 as n, m →∞. (2.7.16)

The relation (2.7.16), Propositions 2.7.1 and 2.7.2 imply that there exist a ˚ subsequence {nk | k ∈ N} and a δ-exceptional set B andanelementM ∈ M such that

L(Pi)(Ω0)=1 for all i ∈ S0 − B, (2.7.17)

◦ (δ) ◦ [u ] = { ∈ | ( ( )) nk ( ) where Ω0 ω Ω unk X ω,s and M ω,s converge uniformly ◦ (δ) ◦ to u(X (ω,s)) and M(ω,s) on each S-bounded subset of Tδ, respectively}. Moreover, we have ◦e(M [un]−M) −→ 0,n−→ ∞ .Actually,wemaytakesome K ∈ ∗N − N. Then from the proof of Proposition 2.7.2,wemaytakeM = [u ] ∗ nK M ,whereunK ∈{un | n ∈ N} is an internal extension of {un | n ∈ N}. Since

[u] [u] [u ] e(M − M)=e(M − M nK ) ≈ 0

[u ] [u] from (2.6.5) in Sect. 2.6,wemayreplaceM = M nK by M = M .

∞ Set A = B ∪ (∪n=1Bn). Since

[u ] (δ) (δ) [u ] nk ( )= ( ( )) − ( ( 0)) − nk ( ) N ω,t unk X ω,t unk X ω, M ω,t , 118 2 Potential Theory we know from (2.7.15)and(2.7.17)that [u] L(Pi) ωN (ω,·) is S-continuous =1, ∀i ∈ S0 − A.

On the other hand, we have N [u](ω,t)=A[u−uk ](ω,t) − M [u](ω,t) − M [un](ω,t) + N [un](ω,t).

Hence, we have

◦ ◦e(N [u]) ≤ 3e(A[u−un])+3e(M [u] − M [un]) .

By letting n −→ ∞ ,weknowfrom(2.6.5) in Sect. 2.6 that e(N [u]) ≈ 0.

2.7.2 Decomposition Under the Whole Measure P

In the following, we shall consider a similar decomposition as in Theorem 2.7.1 under the whole measure P . Just as Lemma 2.7.1,wehave Lemma 2.7.5. For δ ∈ T ,letE(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. For all u ∈ H, t ∈ Tδ and ε>0, we have 2 t/δ t (δ) 2C (1 + δ) (δ) P ω∃s ∈ Tδ (|u(X (ω,s))|≥ε) ≤ E1 (u, u). ε2

Proof. Let A = {i ∈ S0 | u(i) ≥ ε}, and

(δ) (δ) σA (ω)=min{t ∈ Tδ | X (t) ∈ A}.

Then, we have (δ) P ω∃s ∈ Tδ,s≤ t u(X (ω,s)) ≥ ε (δ) −σA /δ −t/δ = Pi ω (1 + δ) ≥ (1 + δ) dm(i) S 0 −σ(δ) /δ t/δ ≤ Ei (1 + δ) A (1 + δ) dm(i) S0 t/δ (δ) =(1+δ) e1 (A)(i) dm(i) S0 t/δ (δ) (δ) (δ) ≤ (1 + δ) E1 e1 (A),e1 (A) , 2.7 Fukushima’s Decomposition Theorem 119 where the reason for the last step holding is (2.3.1) in Sect. 2.3.SinceE(δ)(·, ·) has the Markov property, we get from Corollary 2.2.3 that (δ) (δ) (δ) (δ) |u| |u| E e (A),e (A) ≤ C2E , 1 1 1 1 ε ε 2 C (δ) ≤ E (u, u). ε2 1 Hence, we have 2 t/δ t (δ) C (1 + δ) (δ) P ω∃s ∈ Tδ (u(X (ω,s)) ≥ ε) ≤ E1 (u, u). ε2

Applying the same argument to −u, the lemma follows.

Corollary 2.7.1. For δ ∈ T ,letE(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C.Letu, un,n ∈ N be elements in H, and assume that

◦ (δ) E1 (u − un,u− un) −→ 0 as n −→ ∞ .

{ } ◦ ( (δ)( ·)) There is a subsequence unk such that for almost every ω, unk X ω, ◦ (δ) converges uniformly to u(X (ω,·)) on all S-bounded subsets of Tδ.

Proof. The result follows from Lemma 2.7.5 and basic measure theory. (δ) Deﬁnition 2.7.3. A martingale M with respect to (Ω, Ft ,P) is called 2 ◦ 2 ﬁn a λ -martingale if E(Mt ) < ∞ for all t ∈ Tδ .

Theorem 2.7.2. For inﬁnitesimal δ ∈ T ,letE(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. For any u ∈D(E(δ)), there are two δ-internal additive functionals M [u](ω,t) and N [u](ω,t) such that

[u] [u] [u] (i) A (ω,t)=M (ω,t)+N (ω,t). [u] 2 (δ) (ii) M is a λ -martingale with respect to (Ω, Ft ,P). (iii) N [u](ω,·) is S-continuous for almost all paths ω ∈ Ω under L(P ),and [u] ﬁn E[N ](t) ≈ 0 for all t ∈ Tδ .

Proof. The proof of this result is similar to the one of Theorem 2.7.1.We also refer the reader to the detailed proof of the corresponding result in the symmetric case given in Albeverio et al. [25]. 120 2 Potential Theory 2.8 Internal Multiplicative Functionals

2.8.1 Internal multiplicative functionals

Deﬁnition 2.8.1. For δ ∈ T, an internal function M(ω,t),t∈ Tδ,ω∈ Ω, is said to be a δ-internal multiplicative functional (abbreviated by δ-IMF) of X(δ)(ω,t) if and only if (δ) (i) For each t ∈ Tδ, Mt(·) is Ft -measurable. ∗ (ii) For each t ∈ Tδ,ω∈ Ω, we have M(ω,t) ∈ [0, 1]. (iii) For each ω ∈ Ω, we have

M(ω,0) = 1, (δ) M(ω,t + s)=M(ω,s)M(θs ω,t), ∀t, s ∈ Tδ.

Remark 2.8.1. Let A(ω,t) be a nonnegative δ-internal additive functional. We can deﬁne a δ-IMF M(ω,t) by

M(ω,t)=exp(−A(ω,t)) for all ω ∈ Ω,t ∈ Tδ.

If M(ω,t) is a δ-internal multiplicative functional, let us deﬁne a family t of operators {P | t ∈ Tδ} on H by t (δ) P f(i)=Ei f(X (ω,t))M(ω,t) .

Then, we have for all t, s ∈ Tδ,i∈ S, t+s (δ) P f(i)=Ei f(X (ω,t + s))M(ω,t + s) (δ) (δ) (δ) = Ei f(X (θs ω,t))M(ω,s)M(θs ω,t) (δ) = Ei M(ω,s)EX(δ)(ω,s)[f(X (t))Mt] = P sP tf(i). (2.8.1)

t Hence, {P | t ∈ Tδ} is a semigroup. We call it the semigroup generated by (X(δ),M). Moreover, we have for all f ∈ H,

0 (δ) P f(i)=Ei[f(X (ω,0))M(ω,0)] = f(i).

In particular, we have

P 01=1. (2.8.2) 2.8 Internal Multiplicative Functionals 121

(δ) Since M(ω,δ) is Fδ -measurable, we have

N M(ω,δ)= 1[ˆω](ij)(ω)Mij , i,j=0 where [ˆω](ij)={ω | ω(0) = i and ω(δ)=j} and {Mij | i, j =0, 1, 2,...,N} ∗ is a family of positive hyperreals. Moreover, Mij ∈ [0, 1]. Therefore, the (δ) t transition matrix {pij | i, j =0, 1, 2,...,N} of {P | t ∈ Tδ} is given by

(δ) (δ) ∗ pij = qij Mij ,Mij ∈ [0, 1],i, j =0, 1, 2,...,N. (2.8.3)

From the relation (2.8.3), we know that for all nonnegative internal functions f ∈ H,

t t P f(i) ≤ Q f(i), ∀i ∈ S, t ∈ Tδ. (2.8.4)

2.8.2 Subordinate Semigroups

t Deﬁnition 2.8.2. Asemigroup{P | t ∈ Tδ} of positive linear operators t t from H to H is said to be subordinate to {Q | t ∈ Tδ} if and only if P f(i) ≤ t 0 Q f(i) for all t ∈ Tδ, f ∈ H, f(i) ≥ 0,i∈ S and P = I.

t Theorem 2.8.1. Let {P | t ∈ Tδ} be a semigroup on H. Then the following two conditions are equivalent: t t (1) {P | t ∈ Tδ} is subordinate to {Q | t ∈ Tδ}. (δ) t (2) There exists a δ-IMF M(ω,t) of X (ω,t) generating {P | t ∈ Tδ}.

Proof. (2) =⇒ (1). It follows from the relations (2.8.2)and(2.8.4).

(δ) (1) =⇒ (2). Let {pij | i, j =0, 1, 2,...,N} be the transition matrix of the t semigroup {P | t ∈ Tδ}. Then for each f ∈ H,wehave

N δ (δ) Q f(i)= qij f(j), j=1 N δ (δ) P f(i)= pij f(j). j=1

t t Since {P | t ∈ Tδ} is subordinate to {Q | t ∈ Tδ}, we see that

(δ) (δ) pij ≤ qij , ∀i, j =0, 1, 2,...,N. 122 2 Potential Theory

Deﬁne

(δ) ˆ pij M(i, j)= 1 (δ) , (δ) (qij =0) qij

a ∗ whereweset 0 =0,a∈ [0, ∞). Put

( 0) = 1 M ω, , M(ω,δ)=Mˆ X(δ)(ω,0),X(δ)(ω,δ) . (2.8.5)

For all i ∈ S, we have for any f ∈ H,

N (δ) ˆ (δ) Ei[f(X (ω,δ))M(ω,δ)] = f(j)M(i, j)qij j=1 = P δf(i).

By using mathematical induction, we deﬁne

(δ) ∗ M(ω,(k +1)δ)=M(ω,kδ)M(θkδ ω,δ) for all k ∈ N.

t It is then easy to show that M(ω,t) is a δ-IMF generating {P | t ∈ Tδ}.

2.8.3 Subprocesses

In Sect. 1.5 of Chap. 1, we defined a hyperfinite Markov chain X(δ)(ω,t) (δ) associated with the hyperfinite Dirichlet form E (·, ·). Let us denote (δ) (δ) (δ) Ω, F t ,Y (ω,t),θt ,Pi a hyperfinite Markov chain with state t∈Tδ (δ) (δ) space (S, S).WecallY (ω,t) a subprocess of X (ω,t) if and only if the t (δ) t semigroup P | t ∈ Tδ of Y (ω,t) is subordinate to {Q | t ∈ Tδ} .

Let Y (δ)(ω,t) be a subprocess of X(δ)(ω,t).FromTheorem2.8.1,there exists a δ-IMF M(ω,t) of X(δ)(ω,t) such that

( (δ)( )) = [ ( (δ)( )) ( )] ∈ EP i f Y ω,t Ei f X ω,t M ω,t for all t Tδ, (2.8.6)

(·) (δ)( ) where EP i is the expectation operator corresponding to Y ω,t . We are interested in the following question. Given a δ-IMF M(ω,t),could we construct a subprocess Y (δ)(ω,t) such that the relation (2.8.6)holds? (δ) (δ) (δ) (δ) The answer is yes! In fact, let Ω = Ω, F t = Ft ,Y (ω,t)=X (ω,t), 2.8 Internal Multiplicative Functionals 123

(δ) (δ) t θt = θt . Furthermore, let {P | t ∈ Tδ} be the semigroup given by the { (δ) | =01 2 } relation (2.8.1), and let pij i, j , , ,... be the transition matrix of t ∗ {P | t ∈ Tδ}. Deﬁne for ω ∈ Ω = Ω,k ∈ N,

k−1 P i ([ω]kδ)=δiω(0) p ω(nδ),ω((n +1)δ) . n=0

It is obvious that the relation (2.8.6) holds with respect to Y (δ)(ω,t). We call Y (δ)(ω,t) the canonical subprocess associated with (X(δ),M). We notice { (δ) | =0 1 2 } that pij i, j , , ,...,N need not to have the regularities (1.5.1)and (1.5.2) in Sect. 1.5 of Chap. 1.

2.8.4 Feynman-Kac Formulae

(δ) Let {qi | i =1, 2,...,N} be a 1 × N matrix satisfying

(δ) (δ) 0 ≤ qi ≤ qii ,i=1, 2,...,N.

(δ) (δ) Deﬁne a transition matrix P = {pij | i, j =0, 1, 2,...,N} by

(δ) (δ) (δ) pij = qij − qi δij for i, j =1, 2,...,N, (δ) (δ) (δ) pi0 = qi0 + qi for i =1, 2,...,N, and (δ) (δ) p00 =1,pi0 =0 for i =1, 2,...,N.

(δ) (δ) (δ) pîj =ˆqij − qi δij for i, j =1, 2,...,N, (δ) (δ) (δ) pî0 =ˆqi0 + qi for i =1, 2,...,N and (δ) (δ) pˆ00 =1, pî0 =0 for i =1, 2,...,N.

t (δ) (δ) Let {P | t ∈ Tδ} be the semigroup with P = {pij | i, j =0, 1, 2,...,N} ˆt ˆ(δ) as its transition matrix, and let {P | t ∈ Tδ} be the semigroup with P = (δ) {pˆij | i, j =0, 1, 2,...,N} as its transition matrix. Then, we have 124 2 Potential Theory

Theorem 2.8.2. We have t ˆt (i) The semigroups {P | t ∈ Tδ} and {P | t ∈ Tδ} are dual with respect to m. (ii) The Dirichlet form associated with P (δ) and m is given by

N (δ) (δ) (δ) E (u, v)=E (u, v)+ u(i)v(i)qi m(i). i=1

(iii) There exists a δ-IMF M(ω,t) such that for any f ∈ H, i ∈ S, t ∈ Tδ,

t (δ) P f(i)=Ei[f(X (ω,t))M(ω,t)].

Proof. (i) and (ii) are obvious. (iii) follows easily from Theorem 2.8.1.

2.9 Alternative Expression of Hyperﬁnite Dirichlet Forms

Suppose that μ(·) is a positive internal measure on S. In this section, we shall find conditions on μ and μ-dual transition matrices P = {pij | i, j = ˆ 0, 1,...,N} and P = {pîj | i, j =0, 1, 2,...,N} with the regularities (1.5.1) and (1.5.2) in Sect. 1.5 such that E(·, ·) is the hyperfinite Dirichlet form associated with m and Q.

First we observe that if μ and P and Pˆ have been found, then for all u, v ∈ H, we have ⎡ ⎤ N N 1 ⎣ ⎦ E(u, v)= u(i)v(i)μ(i) − u(j)v(i)pij μ(i) (2.9.1) Δt i=1 j=1 and ⎡ ⎤ N N 1 ⎣ ⎦ E(u, v)= u(i)v(i)μ(i) − u(j)v(i)ˆpjiμ(j) . (2.9.2) Δt i=1 j=1

Therefore, we have from the expression (1.5.19) in Sect. 1.5 and the expression (2.9.1)that

m(i)[1 − qii]=μ(i)[1 − pii] for all i =1, 2,...,N, (2.9.3) m(i)qij = μ(i)pij for all i, j ∈{1, 2,...,N} ,i= j. (2.9.4) 2.10 Transformations of Symmetric Dirichlet Forms 125

Similarly, we have

m(i)[1 − qˆii]=μ(i)[1 − pˆii] for all i =1, 2,...,N, (2.9.5) m(i)ˆqij = μ(i)ˆpij for all i, j ∈{1, 2,...,N} ,i= j. (2.9.6)

On the other hand, if the pairs of (P, μ) and (Q, m) satisfy the conditions (2.9.3)and(2.9.4) (or the pairs of (P,μˆ ) and (Q,ˆ m) satisfy the conditions (2.9.5)and(2.9.6)), then the expression (2.9.1)(or(2.9.2)) holds also. Hence, we get

Theorem 2.9.1. Let μ(·) be a positive internal measure on S,andletP and Pˆ be μ-dual transition matrices. Then the conditions (2.9.3), (2.9.4), (2.9.5), and (2.9.6) are suﬃcient and necessary conditions such that the expression (1.5.19)inSect.1.5 and the expressions (2.9.1)and(2.9.2) hold.

2.10 Transformations of Symmetric Dirichlet Forms

In this section, we assume that

qii =0for all i =1, 2,...,N. (2.10.1)

In fact, this assumption will not affect our theory. The reason comes from the proof of Proposition 1.5.1. Actually, for the general hyperfinite quadratic form E(·, ·) of the expression (1.5.19) in Sect. 1.5, we define

m˜ (i)=(1− qii)m(i), q˜ii =0 for all i =1, 2,...,N.

Moreover, if i =1, 2,...,N and qii < 1, deﬁne

qij q˜ij = for j = i, j ∈{0, 1, 2,...,N} ; 1 − qii if i =1, 2,...,N and qii =1, deﬁne

q˜ij =0 for j =0,i and q˜i0 =1.

Besides, let q˜00 =1, q˜0j =0,j =1, 2,...,N. It is very easy to verify from the Beurling–Deny formulae, Lemma 1.5.1 that the hyperﬁnite quadratic form ˜ associated with m˜ and Q = {q˜ij | i, j =0, 1, 2,...,N} is E(·, ·). Since

qiim(i)=ˆqiim(i), ∀i =1, 2,...,N, 126 2 Potential Theory we may suppose qˆii =0, ∀i =1, 2,...,N. Hence, we may assume 1 = ( +ˆ ) qii 2 qii qii =0,i=1, 2,...,N.

Let Φ be an internal nonnegative function in H. We deﬁne the following quadratic form Φ 1 E (u, v)= u(i) − u(j) v(i) − v(j) Φ(i)Φ(j)qij m(i). Δt 1≤i

Φ Φ It is easy to see that E (·, ·) satisﬁes Proposition 1.5.1 (iii). Moreover, E (·, ·) is symmetric. Thus by Proposition 1.5.1, there exists a transition matrix = { | 0 ≤ ≤ } (·) P pij i, j N and a symmetric measure m such that Φ Δt 1 E (u, v)= u(i) v(i) − P v(i) dm(i). (2.10.3) S0 Δt

In the following, we will ﬁnd the P and m. From the expression (2.10.3)and the Beurling–Deny formulae (Lemma 1.5.1), we have 1 E Φ( )= ( ) − ( ) ( ) − ( ) ( ) u, v u i u j v i v j pij m i Δt 1≤i

Comparing the expressions (2.10.2)and(2.10.4), we must have

( ) ( ) ( )= ( ) 1 ≤ ≤ Φ i Φ j qij m i pij m i for all i, j N, (2.10.5) ( )=0 =1 2 pi0m i for all i , ,...,N.

Therefore, we get

N ( )= ( ) m i pij m i j=0 = ( ) ( ) ( )+ ( ) Φ j qij Φ i m i piim i j =i = [ ( ( ))] ( ) ( )+ ( ) Ei Φ X Δt Φ i m i piim i . (2.10.6) 2.10 Transformations of Symmetric Dirichlet Forms 127

Deﬁne

=1 p00 , =0 =1 2 p0i for i , ,...,N, =0 =1 2 pii for i , ,...,N.

Then from (2.10.6), we obtain

m(i)=Ei[Φ(X(Δt))]Φ(i)m(i).

For i =1, 2,...,N,if EiΦ(X(Δt)) =0, we see from the relation (2.10.5)that for all j =1, 2,...,N,j = i,

q Φ(j) = ij pij Ei[Φ(X(Δt))]

qij Φ(j) = . (2.10.7) l =i qilΦ(l)

=1 2 [ ( ( ))] = 0 1 ≤ ≤ = For i , ,...,N,ifEi Φ X Δt , we can deﬁne pij , j N,j i arbitrarily such that

N =1 pij . (2.10.8) j=1

=0 =0 From (2.10.7)and(2.10.8), we get pi0 for all i . In the following discussion, we suppose that

Ei[Φ(X(Δt))] > 0 for all i =1, 2,...,N. (2.10.9)

Hence, we have from (2.10.7) that for all f ∈ H,

Ei[(fΦ)(X(Δt))] P f(i)= . Ei[Φ(X(Δt))]

Let {P t | t ∈ T } be the semigroup generated by P .Thenitiseasytosee that {P t | t ∈ T } is subordinate to {Qt | t ∈ T }.ByusingTheorem2.8.1, there exists a Δt-IMF M(ω,t) of X(ω,t) generating {P t | t ∈ T }. From the relation (2.8.5) in Sect. 2.8, we know that

Φ(X(ω,Δt)) M(ω,Δt)= . EX(ω,0)[Φ(X((Δt))] 128 2 Potential Theory

Therefore, we have for all k ∈ ∗N,

k Φ(ω,(l +1)Δt) M(ω,(k +1)Δt)= . (2.10.10) [ ( ( ))] l=0 EX(ω,lΔt) Φ X Δt

On the other hand, let Φ be an internal function in H satisfying the hypothesis (2.10.9). We deﬁne M(ω,t) directly by the relation (2.10.10). Then, we have the following:

Theorem 2.10.1. Assume that the hypotheses (2.10.1)and(2.10.9) hold. Then Φ (1) The semigroup {(Q )t | t ∈ T } generated by M(ω,t) is symmetric with respect to the measure Ei[Φ(X(Δt))]Φ(i)m(i)=m(i). Φ Φ (2) E (·, ·) is the hyperﬁnite Dirichlet form associated with {(Q )t | t ∈ T } and m(·).

Proof. The proof is straightforward. Chapter 3 Standard Representation Theory

The purpose of this chapter is to study the standard projection of hyperfinite Markov chains, and the standard projection of hyperfinite Markov chains associated with hyperfinite quadratic forms. At first, we shall introduce in Sect. 3.1 a concept of irregularity; and then we shall prove that if a hyperfinite Markov chain X(·,t) has a set of irregularities, its standard part x(·,t) is a strong Markov process. In Sect. 3.2, we shall find conditions on hyperfinite quadratic forms which guarantee that the modified standard parts of associated hyperfinite Markov chains are strong Markov processes.

The old version of the materials of this chapter can be found in Chap. 5, Albeverio et al. [25]. However, Chap. 5 of Albeverio et al. [25]onlyhan- dles symmetric hyperfinite Dirichlet form. Moreover, many restrictions in Albeverio et al. [25] are not necessary. For instance, we simplify the definition of exceptional sets in Chap. 2 and remove the unnecessary assumptions. Using the updated version of exceptional sets, we will define a concept of exceptional irregularities in Sect. 3.1. We will show that a hyperfinite Markov chain with exceptional irregularities has a standard part, which is a strong Markov process.

In Sect. 3.2, we shall impose conditions directly on hyperfinite quadratic forms to achieve our goal. Again, many assumptions of Albeverio et al. [25] will be simplified. For instance, the separation of compacts of Albeverio et al. [25] will be replaced by separation of points; all concepts involving exceptional sets will be updated using our definition in Chap. 2. It is our hope that the materials of this chapter update and significantly improve the related materials of Chap. 5, Albeverio et al. [25].

One may want to notice that we will impose conditions directly on a standard Dirichlet form in Chap. 4 to get related strong Markov processes. The materials of Chaps. 1, 2,and3 can be thought as the theory of hyperﬁnite Dirichlet forms and hyperﬁnite Markov chains. With the solid work of these three chapters, we will be able to attack the important construction of strong Markov processes associated with standard Dirichlet forms in Chap. 4.

S. Albeverio et al., Hyperﬁnite Dirichlet Forms and Stochastic Processes, 129 Lecture Notes of the Unione Matematica Italiana 10, DOI 10.1007/978-3-642-19659-1_3, c Springer-Verlag Berlin Heidelberg 2011 130 3 Standard Representation Theory 3.1 Standard Parts of Hyperﬁnite Markov Chains

In this section, we shall study standard parts of hyperfinite Markov chains. In order to take standard parts we need a topology. We shall assume that with the exception of the trap s0, the state space S is embedded in the nonstandard version ∗Y of some Hausdorff space Y .IfX is a hyperfinite Markov chain taking values in S, we want to find conditions that guarantee that the standard part of X exists and is a Y -valued Markov process. It turns out that there are two difficulties we shall have to overcome. The first is that the paths of X may be so irregular that no natural standard part process exists. The second is that even when a standard part does exist, there is no reason why it should automatically be a Markov process – taking standard parts we may lump together states that should be kept apart. By using the theory of right standard parts developed by Albeverio et al. [25], Chap. 4, we can solve the first of these problems. Thus, most of our work will be directed to the second problem under discussion. Before we delve into the technicalities, we shall discuss the problem informally in some more details.

Assume that x : Ω × R+ −→ Y is the standard part of X. We would like to prove that x is a Markov process with respect to the filtration it generates. Given that x(t)=y, in general there will be several states s ∈ S0 such that y =st(s), and X may be in any one of them. From the nonstandard point of view, these states are totally unrelated, and hence the past and the future of the process may differ widely from one state to the next. Observation of the whole past may indicate which states are more likely to occur, and thus influence our prediction of the future. This explains why in general x is not a Markov process.

Note, however, that if the process started at si and the process started at sj have the “same” future whenever si ≈ sj, the above argument breaks down, and it is reasonable to expect that x is Markov. One way of formulating this condition is to demand that

L(Pi){ω | x(ω,t) ∈ B} = L(Pj){ω | x(ω,t) ∈ B} (3.1.1) for all t ∈ R+ and all Borel sets B. But the above relation turns out to be too strict for the applications. Instead of demanding that it holds for all infinitely close si,sj, we shall only demand that it holds for all such si,sj outside an exceptional set (i.e., a set which the process hits with probability zero). It may at first seem that little is achieved by allowing a condition to fail on an exceptional set, but in fact the extra freedom and flexibility we gain will turn out to be very useful.

The present section falls into two halves. In the first part, we develop the necessary theory for the inner standard part of sets and a general theory of 3.1 Standard Parts of Hyperfinite Markov Chains 131 exceptional sets. In the second part, we shall show that the standard part of hyperfinite Markov chain is a strong Markov process under some conditions.

3.1.1 Inner Standard Part of Sets

By a hyperﬁnite Markov chain X in this section we shall understand a stationary Markov chain as described in Sect. 1.5,(1.5.1)–(1.5.7). If X has a dual hyperﬁnite Markov chain Xˆ as introduced in Sect. 1.5, we have for all k ∈ ∗N

N P {ω | X(ω,kΔt)=si} = P {ω | X(ω,0) = sj,X(ω,kΔt)=si} j=0 N (kΔt) = qji m(j) j=0 N (kΔt) = qˆij m(i) j=0 = m(i).

However, we shall NOT assume the duality of X in the following discussion, i.e., we do not need condition (1.5.13) in Sect. 1.5. We shall instead assume that for each i ∈ S0, the function

t → P {ω | X(ω,t)=si} is decreasing.

∗ We shall further assume that S0 ⊂ Y for some topological space Y , but ∗ that the trap s0 is not an element of Y . Similarly as in Sect. 2.1, we define the exceptional sets and properly exceptional sets of X using Definition 2.1.1 and Definition 2.1.2. Furthermore, Lemma 2.1.1 and Proposition 2.1.1 hold for X.

Most of the exceptional sets we encounter in the theory of Markov processes are sets we would like to avoid. From the standard point of view, this means that a point y in Y should be avoided if all its nonstandard represen- −1 tations si ∈ st (y)∩S0 are in the exceptional set. To study this relationship closely, we deﬁne the inner standard part A◦ of a subset A of S by ◦ −1 A = y ∈ Y | st (y) ∩ S0 ⊂ A . 132 3 Standard Representation Theory

The standard part of a set A is deﬁned by

◦A =st(A) = {y ∈ Y |∃s ∈ A(st(s)=y)} .

The inner standard part can also be deﬁned in terms of the standard part operation:

A◦ = C(st(CA)), (3.1.2)

where the outer complement is with respect to Y , and the inner complement is with respect to S0.

It is trivial to check that standard parts commute with arbitrary unions. Using the operation (3.1.2), we get that inner standard parts commute with arbitrary intersections. The other way around is less nice, the standard parts and intersections do not commute, and neither do inner standard parts and unions. All we can say is the following:

Lemma 3.1.1. (1) If {An | n ∈ N} is a decreasing family of internal sets, then we have ◦ ◦ An = An. (3.1.3) n∈N n∈N

(2) If {Bn | n ∈ N} is an increasing family of internal sets, then we have ◦ ◦ Bn = Bn. (3.1.4) n∈N n∈N

Proof. Obviously, the left hand of the property (3.1.3) is contained in the set ontheright.Toprovetheconverse,let ◦ x ∈ An. n∈N

For each n ∈ N,pickyn ∈ An such that x =st(yn). Extend {An | n ∈ N} ∗ to be a decreasing internal family {An | n ∈ N} and {yn | n ∈ N} to be an ∗ ∗ internal sequence {yn | n ∈ N} such that yn ∈ An for all n ∈ N.

For each neighborhood O of x, consider the set

∗ ∗ NO = {n ∈ N | yn ∈ O}. 3.1 Standard Parts of Hyperﬁnite Markov Chains 133

All these sets are internal and contain N. Hence, we can ﬁnd an inﬁnite integer η that is in all of them. But then, we must have

x =st(yη).

∗ Since the family {An | n ∈ N} is decreasing, we have yη ∈ Aη ⊂∩n∈NAn. This proves the property (3.1.3).

We now get the property (3.1.4) from the property (3.1.3)byusingthe operation (3.1.2): ◦ Bn = C st C Bn n∈N n∈N = C st (CBn) n∈N = C (st (CBn)) n∈N = C (st(CBn)) n∈N ◦ = Bn. n∈N

Since in general ◦ ◦ Bm,n = Bm,n, m∈N n∈N m∈N n∈N there is no obvious reason to believe that the inner standard part of a properly exceptional set is always Borel. However, we shall now prove that it must at least be universally measurable.

Lemma 3.1.2. Assume A = ∪m∈N ∩n∈N Bm,n for a family {B}m,n∈N of internal sets. For any completed Borel probability measure μ on Y , the inner ◦ standard part A is μ measurable. Moreover, there exists a family {Dm,n | m, n ∈ N} of internal sets such that A ⊂ Dm,n n∈N m∈N 134 3 Standard Representation Theory and ◦ ◦ μ Dm,n − A =0. n∈N m∈N

Proof. We may assume that the family {Bm,n} is increasing in m and decreasing in n.Notethat A = Bm,n m∈N n∈N = Bm,f(m). f∈NN m∈N

Let f¯(m) be the sequence f(0),f(1), ··· ,f(m), and deﬁne Cf¯(m) = Bk,f(k). k≤m

For ﬁxed f, the sequence {Cf¯(m)}m∈N is increasing. Hence by Lemma 3.1.1 and the fact that inner standard parts and arbitrary intersections commute, we have ◦ ◦ A = Cf¯(m). (3.1.5) f∈NN m∈N

C ◦ =st(C ) Since Cf¯(m) Cf¯(m) is closed, the complement ◦ ◦ CA = CCf¯(m) f∈NN m∈N of A can be derived from the closed sets by the Souslin operation, and hence A◦ is measurable with respect to any completed Borel measure (see [321], page 50, for an easy proof).

It only remains to ﬁnd the family {Dm,n}.From(3.1.5) it follows that for each ε ∈ R+, there is a function fε : N −→ N such that ⎛ ⎞ ⎝ ◦ ◦⎠ μ Cg¯(m) − A <ε,

g≤fε m∈N where g ≤ fε means that g(n) ≤ fε(n) for all n ∈ N. Since our original sequence {Bm,n | n ∈ N} is decreasing, we have 3.1 Standard Parts of Hyperﬁnite Markov Chains 135 ◦ ◦ = ¯ Cg¯(m) Cfε(m). g≤fε m∈N m∈N

= ¯ Putting Dm,n Cf1/n(m), the lemma follows. Remark 3.1.1. The argument above shows that a subset of a Hausdorﬀ space can be derived from the closed sets by using the Souslin operation if and only if it is the standard part of a set derived from the internal sets by the same operation. This result and the proof we have given are due to Henson [201].

We have now reached the last lemma we shall need before we can return to our Markov processes. It will be used to pick hyperﬁnite representations of measures avoiding properly exceptional sets. Lemma 3.1.3. Let D be a subset of S0 of the form D = n∈N m∈N Dm,n for a family {Dm,n} of internal sets. If μ is a Radon probability measure on ◦ Y with μ(D )=0, there exists an internal probability measure ν on S0 such that μ = L(ν) ◦ st−1 and L(ν)(D)=0.

Proof. We may obviously assume that the family {Dm,n} is increasing in m ◦ and decreasing in n. Deﬁne a new measure μm,n by μm,n(B)=μ(B − Dm,n). Then, we have

μ(B)=sup inf μm,n(B). (3.1.6) n∈N m∈N

◦ The set S0 −Dm,n is S-dense in Y −Dm,n. Hence, there is an internal measure νm,n concentrated on S0 − Dm,n such that

−1 μm,n = L(νm,n) ◦ st . (3.1.7)

We may choose these measures such that the family {νm,n} is decreasing in ∗ m and increasing in n. Extending to an internal sequence {νm,n | m, n ∈ N}, we ﬁrst pick a γ ∈ ∗N − N such that

◦ ◦ νγ,n(S0) = lim νm,n(S0) m→∞ for all n ∈ N,andthenanη ∈ ∗N − N such that

◦ ◦ νγ,η(S0) = lim νγ,n(S0). n→∞

By using equations (3.1.6)and(3.1.7), and the deﬁnitions of γ and η,wesee −1 that L(νγ,η)(D)=0and μ = L(νγ,η) ◦ st . The measure νγ,η has all the properties of the desired measure, except that we still have to show that it is a probability measure. However, this is clear since we have νγ,η(S0)=1+ε −1 for some ε ≈ 0. Putting ν =(1+ε) νγ,η, the lemma is then proved. 136 3 Standard Representation Theory

By combining Lemmas 3.1.2 and 3.1.3,weget:

Corollary 3.1.1. Let A ⊂ S0 be a properly exceptional set, and let μ be a completed Borel probability measure on Y such that μ(A◦)=0. Then there −1 is an internal probability measure ν on S0 such that μ = L(ν) ◦ st and L(ν)(A)=0.

3.1.2 Strong Markov Processes and Modiﬁed Standard Parts

Having completed our study of exceptional sets and inner standard part of sets, we now turn to the real subject matter of this section, an investigation of standard parts of hyperﬁnite Markov chains. Let us ﬁrst describe what kind of processes we would like to obtain as standard parts [152].

Let Y be a Hausdorﬀ space and let Δ be an extra element. Set YΔ = Y ∪{Δ} . Consider the topological Borel ﬁeld B(Y ) on Y .LetYΔ have the σ-algebra B(YΔ) generated by the Borel sets on Y and the singleton {Δ}. Our standard processes will be YΔ valued, and the new element Δ will serve as a trap.

Assume that (Ω,Π) is a measurable space and {Πt | t ∈ R+} is a family of sub-σ-algebras of Π satisfying Πs ⊂ Πt for all s ≤ t and Πt = Πs for all t ∈ R+. (3.1.8) s>t

We call {Πt | t ∈ R+} a right continuous ﬁltration on Ω if it satisﬁes the condition (3.1.8). Set Π∞ = σ {Πt | t ∈ R+} . In our situation, we can take Π = Π∞.

Amapσ : Ω → [0, ∞) is called a stopping time of {Πt | t ∈ R+} if for all t

{ω | σ(ω) ≤ t}∈Πt.

For each stopping time σ,weintroduceaσ-algebra Πσ by

Πσ = {A ∈ Π∞ |∀t(A ∩ (σ ≤ t) ∈ Πt)} .

Denote by M(Y ) the set of all positive Radon measures on (Y,B(Y )) with finite masses. Given an element ν ∈M(Y ), the completion of the σ-field ν B(Y ) with respect to ν is denoted by B(Y ) . A set or a function is called ν universally measurable if it is measurable with respect to ν∈M(Y ) B(Y ) . 3.1 Standard Parts of Hyperfinite Markov Chains 137

If for each y ∈ YΔ, we are given a probability measure Θy on (Ω,Π). Then for each ν ∈M(Y ),wehave

Θν (A)= Θy(A) dν(y),A∈ Π, Y provided, of course, that this makes sense, i.e., y → Θy(A) is μ measurable.

Finally, if x(t) is a YΔ valued process, its lifetime ζ is deﬁned by

ζ(ω)=inf{t ≥ 0 | x(ω,t)=Δ} .

Definition 3.1.1. We call (Ω,Π∞, {Πt | t ∈ R+} ,x(t),Θy) a strong Markov process,if{Πt | t ∈ R+} is a right continuous filtration of Π, each Θy is a probability measure on Π∞, and x : Ω ×[0, ∞] −→ YΔ is a stochastic process on (Ω,Π∞,Θy) for each y ∈ YΔ. Moreover, the following conditions are satisfied:

(i) For all t ≥ 0 and all measurable E ⊂ Y, the map y → Θy(x(t) ∈ E) is universally measurable in y ∈ Y. (ii) x(ω,∞)=Δ, ∀ω ∈ Ω. x(ω,t)=Δ, ∀t ≥ ζ(ω). (iii) For each y ∈ YΔ, the process x is adapted to (Ω,{Πt} ,Θy),t → xt(ω) is right continuous from [0, ∞) to YΔ,Θy-a.e.ω and lims↑txs(ω) exists in Y for all t ∈ (0,ζ(ω)),Θy-a.e.ω (a.e. represents almost every). (iv) ΘΔ(x(t)=Δ)=1, ∀t ≥ 0; Θy(x(0) = y)=1, ∀y ∈ Y. (v) For all {Πt} stopping times σ, all measurable E ⊂ YΔ, all μ ∈M(Y ), and all s ∈ R+, we have

Θμ(x(σ + s) ∈ E | Πσ)=Θx(σ)(x(s) ∈ E),Θμ-a.e. (3.1.9)

In order to prove that a class of hyperfinite Markov chains have standard parts that are strong Markov processes, we shall have to overcome two main difficulties: the construction of the family of measures {Θy}, and the proof of the strong Markov property (3.1.9). But first we must introduce the necessary regularity conditions in our nonstandard processes.

Here, we would remind the reader that we use Pi to mean the internal probability measure on (Ω,Ft),i ∈ S, t ∈ T , and we use Θy to mean the standard probability measure on (Ω,Π∞),y ∈ Y. We hope this will not cause confusion.

Deﬁnition 3.1.2. Let f : T −→ ∗R be internal. We say that r ∈ R is the S-right limit of f at t ∈ [0, ∞) if for any standard ε>0, there is a standard δ>0 such that if s ∈ T and t<◦s

∗ (δ) Recall that S0 = S0 ∩ Ns( Y ) and that X = X|Tδ. The lifetime ζδ of X(δ) is deﬁned by ◦ (δ) ζδ(ω)=inf t | X (ω,t) ∈/ S0 .

◦ (δ)+ We deﬁne the right standard part X as follows. If t<ζδ(ω),let

◦X(δ)+(ω,t)=S- lim X(δ)(ω,s) s↓t

(δ) if this limit exists and for all t1

◦X(δ)+(ω,t)=Δ

◦ (δ)+ else. If t ≥ ζδ(ω), we always put X (ω,t)=Δ.

Deﬁnition 3.1.3. A subset A of S0 is called a set of irregularities of X if there is a positive inﬁnitesimal δ0 ∈ T satisfying: (δ ) (i) For all si ∈ S0 − A and L(Pi)-a.e. ω, the path X 0 (ω,·) has S-right

and S-left limits at all t<ζδ0 (ω). (ii) For all si ∈ S0 − A,theset ∃ ∈ ﬁn(◦ ( ) ∧ (δ0)( ) ∈ ) ω t Tδ0 t>ζδ0 ω X ω,t S0

has L(Pi) measure zero. (iii) For all inﬁnitely close si,sj ∈ S0 − A,

◦ (δ0)+ ◦ (δ0)+ L(Pi){ X (ω,t) ∈ B} = L(Pj){ X (ω,t) ∈ B}

for all ﬁnite t ∈ [0, ∞) and all Borel sets B.

The hyperﬁnite Markov chain X has δ0-exceptional irregularities if the set A of irregularities of X in the Deﬁnition 3.1.3 can be taken as a δ0-exceptional set. Hence, X has exceptional irregularities if it has δ0- exceptional irregularities for some δ0 ≈ 0,δ0 ∈ T.

The first condition of the above Definition 3.1.3 guarantees that the hyperfinite Markov chainX has a reasonable standard part. The second says that “infinite” is a trap. The third one is a version of the condition (3.1.1).

Putting st(si)=Δ when si ∈ S − S0, we have the following definition of the modified standard part of X(ω,t). Definition 3.1.4. Assume that X has exceptional irregularities, and let A be a properly δ0-exceptional set of irregularities, where δ0 is as in Definition 3.1.3.Letx : Ω × R+ −→ YΔ be defined by 3.1 Standard Parts of Hyperfinite Markov Chains 139

(i) if X(ω,0) ∈/ A, then x(ω,t)=◦X(δ0)+(ω,t). (ii) if X(ω,0) ∈ A, then x(ω,t)=st(X(ω,0)) for all t ∈ R+.

We call x modiﬁed standard part of X(ω,t).

Our aim is to prove that if X has exceptional irregularities, then with an appropriate definition of the family {Θy} of measures, the modified standard parts of X are strong Markov processes. The first step toward the definition of {Θy} is the following version of condition (iii) in Definition 3.1.3.

If ν is an internal probability measure on S,letPν be the measure on Ω deﬁned by

Pν (C)= Pi(C) dν(si). (3.1.10)

Lemma 3.1.4. For δ0 ≈ 0,δ0 ∈ T ,letA be a properly δ0-exceptional set of irregularities of X,andletx be the corresponding modiﬁed standard part of X.Letν1,ν2 be two internal probability measures on S0 such that L(ν1)(A)= −1 −1 L(ν2)(A)=0and L(ν1) ◦ st = L(ν2) ◦ st , then for all t ∈ R+ and all Borel sets B

L(Pν1 ){x(ω,t) ∈ B} = L(Pν2 ){x(ω,t) ∈ B}.

−1 −1 ˜ Proof. Let μ = L(ν1) ◦ st = L(ν2) ◦ st . Choose t ≈ t so large that

◦ (δ0) ˜ X (ω,t)=x(ω,t),L(Pν1 ) and L(Pν2 )-a.e.ω ∈ Ω and pick an internal set B˜ such that

(δ0) ˜ ˜ ◦ (δ0) ˜ L(Pνi )({X (ω,t) ∈ B}Δ{ X (ω,t) ∈ B})=0 for i =1, 2. Deﬁne a function f : Y −→ R as follows:

◦ (1) If y/∈ A , let f(y)=L(Pi){x(ω,t) ∈ B} for some (then for all) si ∈ −1 st (y) ∩ S0 − A. (2) If y ∈ A◦, deﬁne f(y) arbitrarily.

(δ0) ˜ ˜ The function si → Pi{X (ω,t) ∈ B} is a lifting of f with respect to both ν1 and ν2. Therefore, we have

◦ (δ0) ˜ ˜ L(Pν1 ){x(ω,t) ∈ B} = P ν1 {X (ω,t) ∈ B} ◦ (δ0) ˜ ˜ = Pi{X (ω,t) ∈ B} dν1(si) = f(y) dμ(y) 140 3 Standard Representation Theory ◦ (δ0) ˜ ˜ = Pi{X (ω,t) ∈ B} dν2(si)

◦ (δ0) ˜ ˜ = P ν2 {X (ω,t) ∈ B} = ( ){ ( ) ∈ } L P ν2 x ω,t B , and the lemma is proved.

Lemma 3.1.5. For δ0 ≈ 0,δ0 ∈ T , assume that X has a properly δ0- exceptional set A of irregularities, and let x be the modified standard part. For all infinitely close si,sj ∈ S0 − A, all finite sequences t1

Proof. We shall prove this by induction on the length n of the sequences t1

◦ (δ0) ˜ X (ω,tl)=x(ω,tl),L(Pk)-a.e. ˜ ˜ ˜ for l =1, 2, ··· ,nand k = i, j. Choose internal sets B1, B2, ··· , Bn such that (δ0) ˜ ˜ ◦ (δ0) ˜ L(Pk) {X (ω,tl) ∈ Bl}Δ{ X (ω,tl) ∈ Bl} =0 for l =1, 2, ··· ,n and k = i, j.

We deﬁne two measures νi,νj on S by putting

(δ0) ˜ (δ0) ˜ ˜ νk(s)=Pk{ω | X (ω,tn−1)=s and for all l

−1 −1 L(νi) ◦ st = L(νj) ◦ st , and since A is properly δ0-exceptional, we have L(νi)(A)=L(νj)(A)=0. Applying Lemma 3.1.4 with t = tn − tn−1,weget

L(Pνi ){x(tn − tn−1) ∈ Bn} = L(Pνj ){x(tn − tn−1) ∈ Bn}.

Since X is Markov and time homogeneous, the lemma follows.

In the following, we will deﬁne {Πt} and {Θy} of our standard Markov ◦ processes. For t ∈ R+,letΠt be the σ-algebra generated by the sets 3.1 Standard Parts of Hyperﬁnite Markov Chains 141 ωx(ω,t1) ∈ B1 ∧···∧x(ω,tn) ∈ Bn , where 0 ≤ t1 t

The ﬁltration {Πt | t ∈ [0, ∞)} is right continuous. Let us set

Π∞ = ∨{Πt | t ∈ [0, ∞)} .

It follows from Lemma 3.1.5 that for all C ∈ Π∞ and all inﬁnitely close si,sj in S0 − A

L(Pi)(C)=L(Pj)(C).

This observation makes it possible to give the following deﬁnition of a family {Θy | y ∈ YΔ} of measures on Π∞.

◦ If y/∈ A , let for all C ∈ Π∞

−1 Θy(C)=L(Pi)(C) for all si ∈ st (y) − A.

◦ If y ∈ A ∪{Δ} ,C ∈ Π∞, let 1, if C contains all constant paths x(t)=y, Θy(C)= 0, else.

Observe that since a set C ∈ Π∞ contains either all or none of the constant paths x(ω,t)=y, the set function Θy is a measure.

We have reached our goal:

∗ Theorem 3.1.1. Assume that S0 is a hyperfinite subset of Y for some Hausdorff space. Let X : Ω × T −→ S be a hyperfinite Markov chain with exceptional irregularities. Assume that for each i ∈ S0, the function t → P {ω | X(ω,t)=si} is decreasing. If x is a modified standard part of X, then (Ω,{Πt | t ∈ R+}, {Θy | y ∈ YΔ},x(t)) is a strong Markov process.

Proof. Since we already know that {Πt} is right continuous, all we have to do is to check conditions (i)-(v) of Definition 3.1.1. Obviously, the (ii), (iii) and (iv) of Definition 3.1.1 are satisfied by our construction. We shall prove (i) and (v).

Given a Radon probability measure μ on Y , we deﬁne two new measures μ0 and μ1 on Y by 142 3 Standard Representation Theory

◦ μ0(B)=μ(B − A ), ◦ μ1(B)=μ(B ∩ A ), where A is the properly δ0-exceptional set of irregularities used in the construction of x. By Corollary 3.1.1, we can ﬁnd an internal measure ν on S0 satisfying

−1 μ0 = L(ν) ◦ st , L(ν)(A)=0. (3.1.11)

We first prove that (i) in Definition 3.1.1 is satisfied:

(i) Given α ∈ [0, 1],aBorelsetE ⊂ Y, and t ∈ R+, we must show that

{y ∈ Y | Θy(x(t) ∈ E) >α} is μ-measurable for all ﬁnite Radon measures μ.Since ◦ ◦ A ∩ E, α ∈ [0, 1) {y ∈ A | Θy {x(t) ∈ E} >α} = ∅,α=1 is universally measurable by Lemma 3.1.2, it suﬃces to show that

◦ {y/∈ A | Θy(x(t) ∈ E) >α} (3.1.12) is μ-measurable. In fact, we only have to show that the set deﬁned by (3.1.12) is μ0 measurable by the deﬁnition of μ0.

Since X(δ0) has S-right limits a.e. with respect to the probability measure ˆ Pν constructed in (3.1.11)byusingtherelation(3.1.10), we can choose t ≈ t so large that ◦ (δ0) ˜ L(Pν )({x(t) ∈ E}Δ{ X (t) ∈ E})=0. There must be an internal set E˜ such that

(δ0) ˜ −1 ˜ L(Pν ){X (t) ∈ (st (E)ΔE)} =0.

Hence, we have

◦ (δ0) ˜ ˜ L(Pi){x(t) ∈ E} = Pi{X (t) ∈ E} for L(ν)-a.e.si. Combining this with the definition of Θy,weseethat (δ0) ˜ ˜ y → Θy{x(t) ∈ E} has i → Pi{X (t) ∈ E} as a ν-lifting. Hence, it is a μ0 measurable function. This proves that the set defined by (3.1.12)isμ0 measurable, and Definition 3.1.1 (i) follows. 3.1 Standard Parts of Hyperfinite Markov Chains 143

(v) we must show that for all {Πt} stopping times σ, all sets B ∈ Πσ and all s ∈ [0, ∞), the equation

Θμ{ω ∈ B | xσ+s ∈ E} = Θxσ {xs ∈ E} dΘμ (3.1.13) B holds for all Radon probability measure μ on Y and all Borel sets E.

First notice that since the paths of x are constant Θμ1 -a.e., we have

Θμ1 {ω ∈ B | xσ+s ∈ E} = Θxσ {xs ∈ E} dΘμ1 . B

Equation (3.1.13) will hold if we may prove

Θμ0 {ω ∈ B | xσ+s ∈ E} = Θxσ {xs ∈ E} dΘμ0 . (3.1.14) B

In the sequel, we will prove (3.1.14). Let ν be the nonstandard representation of μ0 given in the construction (3.1.11). The hyperﬁnite counterpart of (3.1.14)is (δ ) { ∈ | 0 ∈ } = { (δ0) ∈ } Pν ω C Xτ+s F PX(δ0) Xs F dPν , (3.1.15) C τ where C and F are internal sets and C is measurable in the ∗−algebra generated by the internal stopping time τ.SinceX(δ0) is a time homogeneous Markov chain, (3.1.15) holds. Our plan is to deduce (3.1.14)from(3.1.15).

◦ We ﬁrst pick an internal stopping time τ such that τ = σ, L(Pν )-a.e., and such that there is a τ measurable set C satisfying

L(Pν )(BΔC)=0. (3.1.16)

Let P τ be an internal measure on Ω given by τ P (D)= P (δ0 ) (D) dPν . Xτ

Since A is properly exceptional, we observe that

τ (δ0) L(P ){Xt ∈ A} =0

∈ fin for all t Tδ0 (where δ0 is the infinitesimal used in the construction of x, ˜ ∈ fin ˜ = see Definition 3.1.4). Hence, we can choose an s Tδ0 , s s, such that 144 3 Standard Representation Theory

◦ (δ0) τ Xs˜ = xs,L(P )-a.e. (3.1.17) and

◦ (δ0) Xτ+˜s = xσ+s,L(Pν )-a.e. (3.1.18)

Finally, we pick an internal set F such that

(δ0) −1 L(Pν ){Xτ+˜s ∈ st (E)ΔF } =0, (3.1.19)

τ (δ0) −1 L(P ){Xs˜ ∈ st (E)ΔF } =0. (3.1.20)

We now have

Θμ {ω ∈ B | xσ+s ∈ E} 0

= Θy{ω ∈ B | xσ+s ∈ E} dμ0(y) (by def. of Θμ0 )

= L(Pi){ω ∈ B | xσ+s ∈ E} dL(ν)(si) (by def. of Θy)

= L(Pν ){ω ∈ B | xσ+s ∈ E} (by def. of Pν ) ◦ = Pν {ω ∈ C | Xτ+˜s ∈ F } (by (3.1.16), (3.1.18), (3.1.19)) ◦

= PXτ {Xs˜ ∈ F } dPν (by (3.1.15)) C

= L(PXτ ){xs ∈ E} dL(Pν ) (by (3.1.16), (3.1.17), (3.1.20)) B

= Θxσ {xs ∈ E} dΘμ0 (by def. of ν, τ and {Θy}). B

This proves (3.1.14) and the theorem.

Example 3.1.1. Let Z = {··· , −2, −1, 0, 1, 2, ···} (i.e., set of all the integers). Let η ∈ ∗N − N, and set k ∗ S0 = √ k ∈ Z, |k|≤η . η

Deﬁne transition probabilities qs,s by 1 √1 2 if |s − s | = η , qs,s = 0 otherwise.

If s, s ∈ S0,andifs0 is the trap, let 3.2 Hyperﬁnite Dirichlet Forms and Markov Processes 145 1 √ 2 if s = ± η, qs,s = 0 otherwise.

Let m be an internal measure on S0 given by 1 m(s)=√ η for all s ∈ S0, and let a time-line be k ∗ T = k ∈ N0 . η

The process X is Anderson’s random walk with a uniform initial distribution corresponding to the Lebesgue measure. To prove that the standard part of X is a strong Markov process, we show that the empty set is a set of irregularities of X.SinceXisS-continuous L(Pi)-a.e. for all nearstandard si ∈ S0, the two first conditions in Definition 3.1.3 are obviously satisfied. If si ≈ sj, the paths starting at si look exactly like the paths starting at sj except for an infinitesimal translation. Hence, they induce the same standard paths and the property (iii) in Definition 3.1.3 follows.

3.2 Hyperﬁnite Dirichlet Forms and Markov Processes

In this section, we shall obtain conditions on hyperfinite quadratic forms which guarantee that the modified standard parts of the associated hyperfinite Markov chains are strong Markov processes. The method we shall apply is simple, we just use the relationship between forms and processes established in Sect. 1.5 to translate the conditions of Theorem 3.1.1 into the language of hyperfinite quadratic forms. One by one we shall reformulate the conditions of Definition 3.1.3 in terms of hyperfinite quadratic forms.

In this section, we need a slightly stronger assumption on the state space. Let Y be a regular Hausdorff space (or a T3 space), i.e., for all closed sets F and x ∈ F, there are disjoint open sets O1,O2 such that x ∈ O1, F ⊂ O1. ∗ Suppose that S0 = {s1,s2, ··· ,sN } is a hyperfinite subset of Y and m is a ∗ hyperfinite positive measure on S0, N ∈ N − N.LetH be the linear space ∗ of all internal functions u : S0 −→ R with the inner product

N u, v = u(si)v(si)m(si). i=1 146 3 Standard Representation Theory

∗ Given an infinitesimal Δt ≈ 0, denote by T = {kΔt | k ∈ N0} the hyperfinite time line. Let X and Xˆ be the dual hyperfinite Markov chains introduced in Sect. 1.5 with m as dual measure. That is, we assume that conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14) in Sect. 1.5 are fulfilled. Let E(·, ·) be the hyperfinite quadratic form on H associated with the transition matrix Q and the dual measure m, and Eˆ(·, ·) be the co-form of E(·, ·).

Given x ∈ Y , we recall that the monad1 of x is the set of ∗Y deﬁned by μ(x)= {∗Ox ∈ O and O is open}.

3.2.1 Separation of Points

The following assumption will take care of Definition 3.1.3 (i). In Albeverio et al. [25], 5.5.1. Definition, a definition of separation of compacts was given for symmetric hyperfinite Dirichlet forms. The following definition simplifies the condition since a single point can be viewed as a compact set. Definition 3.2.1. Let Z be a subset of Y . A hyperfinite quadratic form E(·, ·) separates points of Z if there exists a countable family π = {un | n ∈ N} of internal functions such that for all x, y ∈ Z, x = y, there is an element un ∈ π satisfying

◦ ◦ un(s1) = un(s2),s1 ∈ μ(x),s2 ∈ μ(y),si ∈ S0,i=1, 2. (3.2.1)

Moreover, we have

◦ E1(un,un) < ∞, ∀un ∈ π. (3.2.2)

We call π = {un | n ∈ N} a separating family of Z by E(·, ·).

We recall from Sect. 2.1 that if δ ∈ T, the sub-line Tδ is deﬁned by

∗ Tδ = {kδ | k ∈ N0},

(δ) (δ) X is the restriction of X|Tδ.Letζ = ζδ be the lifetime of X , i.e.,

◦ (δ) ζδ(ω)=inf{ t | X (ω,t) ∈ S0},

∗ where S0 = S0 ∩ Ns( Y ). For any subset A ⊂ S0,let (δ) ◦ −1 ◦ τA (ω)=inf{ tX(t) ∈ st (A ) and t ∈ Tδ}.

1 For the general concept of monad, we refer to Albeverio et al. [25]. 3.2 Hyperﬁnite Dirichlet Forms and Markov Processes 147

Lemma 3.2.1. Let Y be a regular Hausdorﬀ space, and let A be a subset of ◦ S0. Suppose that E(·, ·) separates the points of Y − A and π is the separating family. For ω ∈ Ω,ifthepathX(·,ω) fails to have an S-left or S-right limit (Δt) at t<ζΔt(ω) ∧ τA (ω), then so does u(X(ω,·)) for some u ∈ π. (Δt) Proof. Let ω be a ﬁxed point in Ω.Fixt<ζΔt(ω)∧τA (ω),t∈ [0, ∞). Given ◦ a sequence {tn | n ∈ N} from T such that the standard part tn increases strictly to t, we shall prove that the sequence ◦ X(ω,tn)n ∈ N has a cluster point in Y − A◦.

Suppose for a contradiction that ◦ X(ω,tn)n ∈ N has no cluster point in Y .Thenforeachy ∈ Y , there is a neighborhood Oy and an integer ny ∈ N such that

◦ X(ω,tn) ∈/ Oy when n ≥ ny. Since Y is regular, we can ﬁnd a neighborhood Gy of y such that the closure Gy is contained in Oy. Hence, we have

∗ X(ω,tn) ∈/ Gy

∗ when n ≥ ny. Extend {tn | n ∈ N} to be an internal sequence {tn | n ∈ N} of elements of T less than t. Consider the set

∗ ∗ Ay = {n ∈ N | n ≤ ny or X(ω,tn) ∈/ Gn} .

∗ Since Ay is internal and contains N, there is an ηy ∈ N − N such that all η ≤ ηy are elements of Ay. By saturation, there is an inﬁnite η less than all ηy.Butthen,wehave ∗ X(ω,tη) ∈/ Gy for all y. This implies that X(ω,tη) is not nearstandard, contradicting our assumption that t<ζΔt(ω).

◦ Suppose that y ∈ Y is a cluster point of { X(ω,tn)}n∈N.Sincetn ↑ t, (Δt) ◦ t<τA (ω), it is easy to see that y/∈ A . Let x ∈ Y − A◦ ˆ=Z be a cluster point of {◦X(ω,t)} . If x is not the S-left limit of X(ω,·) at t, there must be another sequence {sn | n ∈ N} increasing 148 3 Standard Representation Theory

◦ to t such that x is not a cluster point of { X(ω,sn) | n ∈ N}. Repeating the ◦ argument above, we see that { X(ω,sn) | n ∈ N} must have a cluster point y ∈ Z.Letu ∈ π be the function satisfying the conditions of (3.2.1)and (3.2.2). Obviously, u(X(ω,·)) does not have an S-left limit at time t.This proves the S-left case of the lemma. The S-right limit case can be treated in a similar way.

We shall use Lemma 3.2.1 and Fukushima’s decomposition Theorem 2.7.2 to show that if E(·, ·) separates points, then the associated Markov chain X satisfies Definition 3.1.3 (i).Butwemustgetsomepreparationatfirst.

It is easy to see that r r {ω |∃t ∈ Tδ (X(ω,t) ∈ An)} = {ω |∃t ∈ Tδ (X(ω,t) ∈ An)}, n∈N n∈N (3.2.3) for all sequences {An | n ∈ N} of subsets of S. However, the corresponding formula for intersections is false in general. In particular, it does hold if the sequence is decreasing and consists of internal sets. This is the observation behind the next lemma.

Lemma 3.2.2. Let A ⊂ S0, and assume that there is a family {Bm,n | m, n ∈ N} of internal sets such that A = Bm,n m∈N n∈N and for each m, the sequence {Bm,n | n ∈ N} is decreasing. Then r r {ω |∃t ∈ Tδ (X(ω,t) ∈ A)} = ω |∃t ∈ Tδ X(ω,t) ∈ Bm,n . m∈N n∈N (3.2.4)

Proof. It suﬃces to prove that r r ω∃t ∈ Tδ X(ω,t) ∈ Bm,n = ω∃t ∈ Tδ X(ω,t) ∈ Bm,n n∈N n∈N (3.2.5) for all m,since(3.2.4) then follows from (3.2.3). Also, it is immediately clear that the left hand of (3.2.5) is included in the right hand side. To prove the opposite inclusion, choose 3.2 Hyperﬁnite Dirichlet Forms and Markov Processes 149 r ω0 ∈ {ω |∃t ∈ Tδ (X(ω,t) ∈ Bm,n)}. n∈N

Consider the set

∗ r ˜ {n ∈ N |∃t ∈ Tδ (X(ω0,t) ∈ Bm,n)},

˜ ∗ where {Bm,n | n∈ N} is some internal, decreasing extension of {Bm,n | n ∈ N}. By the choice of ω0, this set contains N. Since it is internal, it must have an inﬁnite member η.Thus,wehave r ˜ r ω0 ∈{ω |∃t ∈ Tδ (X(ω,t) ∈ Bm,η)}⊂{ω |∃t ∈ Tδ (X(ω,t) ∈ Bm,n)}. n∈N

The lemma is proved. We are now in the position to prove the following: Proposition 3.2.1. Let Y be a regular Hausdorff space, and let E(δ)(·, ·) be a hyperfinite weak coercive quadratic form for every infinitesimal δ ∈ T . Assume that A is an exceptional set and E(·, ·) separates the points of Y −A◦.

There exists an infinitesimal δ0 ∈ T such that for all infinitesimal δ ∈ Tδ0 , there is a δ-exceptional set A0(δ) satisfying that for all si ∈ S0 − A0(δ), the hyperfinite Markov chain X(δ)(t) has S-left and S-right limits at all t< ζδ,L(Pi)-a.e.

Proof. To find δ0, we look at the property of the separating family π = (δ) (δ) {un | n ∈ N}. From Theorem 1.4.2, we know that D(E )=D(E ) for all ◦ δ ∈ T,δ ≈ 0.Foreachu ∈ π,wehave E1(u, u) < ∞. Hence, we can find (δ) (δ) a δu ≈ 0 such that u ∈D(E )=D(E ) for all infinitesimal δ ≥ δu by Corollary 1.2.4. By saturation, there is a δ0 ≈ 0 larger than all δu and A is δ0-exceptional.

Turning to A0(δ) for infinitesimal δ ∈ Tδ0 ,wefirstobservethatby Lemma 3.2.1 and the countability of π, it suffices to show that for each un ∈ π = {un | n ∈ N},thereisaδ-exceptional set An(δ) such that for all (δ) si ∈ S0 − An(δ), the process un(X (t)) has S-left and S-right limits at all t<∞,L(Pi)-a.e. Actually, we define ∞ A0(δ)=A0 An(δ) , n=1 where A0 is a properly δ-exceptional set containing A. Then, A0(δ) is δ- (δ) exceptional by Lemma 2.1.1 and for all si ∈ Si − A0(δ),un ∈ π, un(X (t)) has S-right and S-left limits for all t<∞,L(Pi)-a.e. Therefore, the hyper- (δ) finite Markov chain X (t) has S-left and S-right limits at all t<ζδ(ω), 150 3 Standard Representation Theory

L(Pi)-a.e. by Lemma 3.2.1. From this observation, we only need to ﬁnd the exceptional set An(δ) for every un ∈ π. For the simplicity of notation, we write u = un in the following.

Given two standard rationals p, q, −∞

0 (δ) τ(p,q) =min{t ∈ Tδ | u(X )(ω,t)) ≤ p}, 2n 2n−1 (δ) τ(p,q) =min{t ∈ Tδ | t>τ(p,q) (ω) ∧ u(X )(ω,t)) ≤ p}, 2n+1 2n (δ) τ(p,q) =min{t ∈ Tδ | t>τ(p,q)(ω) ∧ u(X )(ω,t)) ≥ q}.

Let B ⊂ S0 be deﬁned by n 1 B = {i | Pi{τ ≤ m}≥ }. (p,q) k (p,q) m∈N k∈N n∈N

(δ) If X fails to have S-left or S-right limits with positive L(Pi) probability, then we have si ∈ B. We must show that B is δ-exceptional.

By Lemma 3.2.2, we have for all r ∈ [0, ∞)

{ |∃ ∈ r( ( ) ∈ )} ω t Tδ X ω,t B r n 1 = ω∃t ∈ Tδ PX(δ) (ω,t){τ(p,q) ≤ m}≥ . k (p,q) m∈N k∈N n∈N

If B is not δ-exceptional, there must be a pair (p, q) of rationals, as well as integers m, k ∈ N,r∈ [0, ∞) and an inﬁnite number η ∈ ∗N such that r η 1 L(P ) ω∃t ∈ Tδ PX(δ) (ω,t){τ(p,q) ≤ m}≥ > 0. k

This implies that with positive probability, u(X(δ)) jumps back and forth between p and q more than η times before time m + r.

Since u ∈D(E(δ)), Fukushima’s decomposition Theorem 2.7.2 tells us that

u(X(δ))=N [u] + M [u], where N [u] is S-continuous L(P )-a.e. and M [u] is a λ2-martingale. If u(X(δ)) jumps η times between p and q before t = m+r, there must be an infinitesimal interval where it jumps back and forth infinitely many times. Since N is S-continuous – and hence almost constant on infinitesimal intervals – most 3.2 Hyperfinite Dirichlet Forms and Markov Processes 151 of this jumping is done by M. Hence the quadratic variation of M is infinite on a set of positive measure, contradicting the fact that it is a λ2-martingale.

We then conclude that B must be δ-exceptional, and the proposition is thus proved.

3.2.2 Nearstandardly Concentrated Forms

Now we are in a position to ﬁnd a condition on E(·, ·) that implies a similar property like (ii) in Deﬁnition 3.1.3. For any internal subset C of S0,weset

HC = {u ∈ H | u(si)=0for all si ∈ S0 − C} .

For each u ∈ H and any internal set C ⊂ S0,δ ∈ T, we deﬁne

(δ) (δ) −σ /δ (δ) (δ) ( )= ( ) − i (1 + ) S0−C ( ) uC i u i E δ u X σS0−C , (3.2.6) where (δ) ( )=min ∈ | (δ)( ) ∈ − σS0−C ω t Tδ X ω,t S0 C .

It is easy to see that

(δ) uC ∈ HC .

Deﬁnition 3.2.2. (1) For δ ∈ T , a hyperﬁnite quadratic form E(·, ·) is said to be δ-nearstandardly concentrated if there exists an increasing sequence {Bn | n ∈ N} of internal subsets of S0 such that:

(i) The set ∪n∈NBn − S0 is δ-exceptional. ◦ (δ) (ii) For each F ∈ H, if E1 (F, F) < ∞,thenwehave

◦E(δ)( − (δ) − (δ) ) −→ 0 −→ ∞ 1 F FBm ,F FBm ,m .

(2) A hyperfinite quadratic form E(·, ·) is said to be nearstandardly concentrated if it is δ-nearstandardly concentrated for some infinitesimal δ ∈ T. Notice that above definition of nearstandardly concentrated is different from that of Albeverio et al. [25], 5.5.4 Definition. Our definition is moti- vated by both Albeverio et al. [25], 5.5.4 Definition and Condition (I) of Theorem 4.1.1 in Chap. 4. 152 3 Standard Representation Theory

Lemma 3.2.3. For δ ∈ T, let E(δ)(·, ·) be a hyperfinite weak coercive quadratic form with continuity constant C. Assume that E(·, ·) is δ-nearstandardly concentrated. Let X be the related hyperfinite Markov ◦ (δ) chain. Then for each F ∈ H with E1 (F, F) < ∞,theset fin (δ) A(F )= si ∈ S0 − BnL(Pi) ω∃t ∈ Tδ F (X (t)) ≈ 0 > 0 m∈N (3.2.7) is δ-exceptional.

Proof. Step 1. For ε>0,set D = si|F (si)| >ε , (δ) (δ) σD (ω)=min t ∈ Tδ | X (ω,t) ∈ D , (δ) (δ) −σD (ω)/δ e1 (D)=Ei (1 + δ) .

From Corollary 2.2.3,wehave

2 (δ) (δ) (δ) C (δ) E e (D),e (D) ≤ E (F, F). 1 1 1 ε2 1 Hence, we have ◦ (δ) (δ) (δ) E1 e1 (D),e1 (D) < ∞.

This implies that ◦E(δ) (δ)( ) − ( (δ)( ))(δ) (δ)( ) − ( (δ)( ))(δ) −→ 0 −→ 0 1 e1 D e1 D Bm ,e1 D e1 D Bm ,m , (3.2.8)

( (δ)( ))(δ) where e1 D Bm is defined in the same way as the definition (3.2.6). Step 2. From the approximation (3.2.8), there exists an increasing sequence {mk}k∈N satisfying (δ) (δ) (δ) (δ) (δ) (δ) (δ) δk = E1 e1 (D) − (e1 (D))B ,e1 (D) − (e1 (D))B mk mk < 2−k. 3.2 Hyperfinite Dirichlet Forms and Markov Processes 153

Set ( )= ∈ − | (δ)( )( )|≥ ∈ N Ak F, ε si S0 Bmk e1 D si δk ,k .

Note that if si ∈ Ak(F, ε), we have (δ) (δ) (δ) |e1 (D)(si) − (e1 (D))B (si)|≥ δk. mk

Hence, we have from Lemma 2.7.5 the following 1 (δ) P ω∃t ∈ Tδ X (ω,t) ∈ Ak(F, ε) 1 (δ) (δ) (δ) (δ) (δ) ≤ P ω ∃t ∈ Tδ |e1 (D)(X (ω,t)) − (e1 (D))B (X (ω,t))|≥ δk mk 2 1 2C (1 + δ) δ (δ) (δ) (δ) (δ) (δ) (δ) (δ) ≤ E1 e1 (D) − (e1 (D))B ,e1 (D) − (e1 (D))B δk mk mk 2 1 =2C (1 + δ) δ δk 2 1 −k+1 ≤ C (1 + δ) δ 2 . (3.2.9)

For each K ∈ N, we have ﬁn (δ) A(F, ε)ˆ= si ∈ S0 − BnL(Pi) ∃t ∈ Tδ |F (X (t))| >ε > 0 m∈N ∞ ⊂ Ak(F, ε). k=K

−k Since δk < 2 , it follows from the relation (3.2.9)that

∞ 1 2 1 P {ω |∃t ∈ Tδ (X(ω,t) ∈ A(F, ε))}≤2C (1 + δ) δ δk k=K 2 1 −K+2 ≤ C (1 + δ) δ 2 .

Hence, A(F, ε) is δ-exceptional.

Step 3. Now it is easy to see that A(F ) is δ-exceptional, since A(F )= 1 m∈N A(F, m ). Proposition 3.2.2. For δ ∈ T, let E(δ)(·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Assume that E(·, ·) is δ-nearstandardly concentrated. Let X be the related hyperﬁnite Markov chain. ◦ (δ) Then for each F ∈ H with E1 (F, F) < ∞,thereexistsaδ-exceptional set 154 3 Standard Representation Theory

A1(F, δ) such that for si ∈ S0 − A1(F, δ) ﬁn ◦ (δ) L(Pi) ω∃t ∈ Tδ t ≥ ζδ(ω) ∧ F (X (t)) ≈ 0 =0.

∗ Proof. Let {Bn | n ∈ N} be an increasing extension of {Bn | n ∈ N},which is the increasing sequence in Deﬁnition 3.2.2 (1). Set B = ∪n∈NBn. Deﬁne (δ) (δ) ∗ τn (ω)=min t ∈ TδX (ω,t) ∈/ Bn ,n∈ N, (δ) ◦ (δ) τ (ω)=inf tX (ω,t) ∈/ B .

It is not hard to check that (δ) ◦ τ (ω)=sup τn(ω)n ∈ N .

Let A1(F, δ) be a properly δ-exceptional set containing B − S0 and the set A(F ),whereA(F ) is defined in Lemma 3.2.3 by the definition (3.2.7). Given ∗ si ∈ S0 − A1(F, δ), we can find an η ∈ N − N such that

(δ) ◦ (δ) τ (ω)= τη (ω),L(Pi)-a.e. (3.2.10)

By the deﬁnition of A1(F, δ),wehaveB − S0 ⊂ A1(F, δ). Since A1(F, δ) is properly δ-exceptional and si ∈/ A1(F, δ),thisimpliesthat

τ(ω)=ζδ(ω),L(Pi)-a.e. (3.2.11)

Combining (3.2.10)and(3.2.11), we have

◦ (δ) τη (ω)=ζδ(ω),L(Pi)-a.e. (3.2.12)

Notice that si ∈/ A1(F, δ), A1(F, δ) contains A(F ), and it is properly δ-exceptional. We have

(δ) L(Pi)(X(ω,τη (ω)) ∈ A(F )) = 0.

Hence, we have by the deﬁnition of A(F ) ﬁn (δ) (δ) L(Pi) ω∃t ∈ Tδ t ≥ τη (ω) ∧ F (X (t)) ≈ 0 =0.

The proposition follows from the relation (3.2.12). 3.2 Hyperﬁnite Dirichlet Forms and Markov Processes 155 3.2.3 Quasi-Continuity

∗ Definition 3.2.3. (1) An internal function F : S0 −→ R is said to be hyperfinite δ-quasi-continuous if there is a hyperfinite δ-exceptional set A ⊂ S0 such that for all infinitely close si,sj ∈ S0 − A, we have F (si) ≈ F (sj).

∗ (2) An internal function F : S0 −→ R is called hyperfinite quasi-continuous if there is an infinitesimal δ such that F is hyperfinite δ-quasi-continuous. Remark 3.2.1. We remark that usually one uses zero capacity sets to define the quasi-continuity in regular Dirichlet spaces (refer to Sect. 4.1 of Chap. 4,or refer to [175]). Nevertheless, we utilize the exceptional sets in the definition of the hyperfinite quasi-continuity. The reason is that we have the equivalence between the concept of exceptional sets and that of zero capacity sets in regular Dirichlet space theory, e.g., Theorem 4.3.1 in [175]). However, we only have Theorem 2.4.1,Theorem2.4.2,andTheorem2.4.3 in Sect. 2.4,from which we understand that the conception of exceptional sets is not equivalent to that of zero capacity sets in hyperfinite Dirichlet space theory. Given a hyperfinite δ-quasi-continuous and S-bounded function F ∈ H, let A be a δ-exceptional set satisfying F (si) ≈ F (sj), si ≈ sj, si,sj ∈ S0 −A. Define ◦F = f : Y −→ R by: ◦ ◦ −1 (i) f(y)= F (si) whenever y/∈ A and si ∈ st (y) − A; (ii) f(y)=0whenever y ∈ A◦. In order to get the condition that makes the process X to satisfy Definition 3.1.3 (iii), we impose the following condition on E(·, ·). Definition 3.2.4. (1) A hyperfinite quadratic form E(·, ·) is said to have a hyperfinite δ-quasi-continuous core if there are a family of countable functions π = {Fn | n ∈ N}⊂H and a δ-exceptional set A such that Fn(si) ≈ Fn(sj ), si ≈ sj,si,sj ∈ S0 − A, ∀n ∈ N,and

◦ ◦ (Y − A ) ∩B(Y ) ⊂ σ{fn = F n | n ∈ N}. (3.2.13)

st{max | ( )|} ∞ ◦E(δ)( ) ∞ ∀ ∈ N Moreover, si∈S0 Fn si < , 1 Fn,Fn < , n .πis called a hyperﬁnite δ-quasi-continuous core of E(·, ·).

(2) A hyperfinite quadratic form E(·, ·) is said to have a hyperfinite quasi- continuous core if it has a hyperfinite δ-quasi-continuous core for some infinitesimal δ ∈ T. Lemma 3.2.4. Let M be a set. Consider a countable subset G of M and a countable collection Λ of maps from M × M into M.Then,thereexistsa countable set L such that

(a) G ⊂ L ⊂ M. (b) λ(L × L) ⊂ L, ∀λ ∈ Λ. 156 3 Standard Representation Theory

Proof. This is proven in Fukushima [175] Lemma 6.1.1.

Lemma 3.2.5. Suppose that E(·, ·) has a hyperﬁnite δ-quasi-continuous core ˆ π = {Fn | n ∈ N}. Then, there exists a countable set πˆ = {Fn | n ∈ N} such that ˆ ˆ ˆ ˆ ˆ ˆ ˆ (1) Fn, Fm ∈ π,ˆ a ∈ Q =⇒|Fn|∈π,ˆ Fn + Fm ∈ π,ˆ FnFm ∈ π,ˆ ˆ ˆ aFn ∈ π,ˆ and Fn ∧ a ∈ π,ˆ ˆ (2) π = {Fn | n ∈ N}⊂πˆ = {Fn | n ∈ N}, (3) every element in πˆ is hyperﬁnite δ-quasi-continuous,

where Q = {a1,a2, ··· ,an, ···} is the set of all rational numbers.

Proof. Let us denote by Qδ(S0) the set of hyperﬁnite δ-quasi-continuous functions on S0.Wedeﬁnethemaps

Λ = {λ−2,λ−1,λ0,λ1,λ2, ···} from [H ∩ Qδ(S0)] × [H ∩ Qδ(S0)] into H ∩ Qδ(S0) by

λ−2(u, v)=|u|, λ−1(u, v)=u + v, λ0(u, v)=uv, λ2i−1(u, v)=aiu, λ2i(u, v)=u ∧ ai,i=1, 2, ···

Applying Lemma 3.2.4 to π and Λ, we get a countable set πˆ satisfying (1), (2), and (3).

Lemma 3.2.6. Assume that the hyperﬁnite quadratic form E(·, ·) has a δ-quasi-continuous core π = {Fn | n ∈ N}. For any two ﬁnite measures ν1 and ν2 on (Y,B(Y )),ifν1 and ν2 satisfy the following condition: ◦ ◦ F (y) ν1(dy)= F (y) ν2(dy) for all F ∈ π,ˆ (3.2.14) Y Y where πˆ is the countable set obtained in the Lemma 3.2.5,thenν1 and ν2 coincide on (Y,σ(◦F | ◦F ∈ πˆ)).

◦ Proof. For any F ∈ π,ˆ a ∈ Q, Fn =[n(F − F ∧ a)] ∧ 1 ∈ πˆ and Fn ↑ 1(◦F>a). This implies that

1(◦F (y)>a) ν1(dy)= 1(◦F (y)>a) ν2(dy) Y Y 3.2 Hyperﬁnite Dirichlet Forms and Markov Processes 157 from the conditions (3.2.14). Now we can prove our Lemma by using the monotone class theorem.

Lemma 3.2.7. If the hyperfinite quadratic form E(·, ·) has a hyperfinite δ- ◦ quasi-continuous core π = {Fn | n ∈ N},thenπ separates the points of Y −A by E(δ)(·, ·),whereA is the δ-exceptional set in the Definition 3.2.4 (1).

Proof. Since Y is a Hausdorﬀ space, the singleton {x} is a closed subset of Y . This implies that {x}∈B(Y ) for any x ∈ Y. For x, y ∈ Y − A◦,letus assume that Fn(s1) ≈ Fn(s2) for every Fn ∈ π, s1 ∈ μ(x),s2 ∈ μ(y).Then we obtain ◦ −1 ◦ {x, y}⊂∩ ( Fn) (( Fn)(x)) | Fn ∈ π .

It follows from the condition (3.2.13) and above observation that ◦ −1 ◦ ∩ ( Fn) (( Fn)(x)) | Fn ∈ π = {x} .

Therefore, we get x = y. This shows that π separates the points of Y − A◦ by E(δ)(·, ·).

Lemma 3.2.8. For δ ∈ T ,letE(δ)(·, ·) be a hyperfinite weak coercive quadratic form with continuity constant C. Suppose that E(·, ·) has a hyperfinite δ-quasi-continuous core π = {Fn | n ∈ N} and is δ-nearstandardly ∗ ˆ fin concentrated. For all η ∈ N − N,si ∈/ A2(δ), Fn ∈ π,ˆ t ∈ Tδ , we have ˆ ( (δ)( )) ( ) ≈ ˆ ( (δ)( ))1 ( ) Fn X ω,t Pi dω Fn X ω,t (t<τ (δ)(ω)) Pi dω , Ω Ω η (3.2.15) where A2(δ) is any properly δ-exceptional subset of S0 containing B −S0 and ˆ ˆ the set ∪n∈NA(Fn). Here, A(Fn) is the set defined in Lemma 3.2.3 by (3.2.7).

ˆ Proof. Notice that si ∈/ A2(δ),andA2(δ) contains ∪n∈NA(Fn) and is properly δ-exceptional. We have

(δ) ˆ L(Pi)(X(ω,τη (ω)) ∈ A(Fn)) = 0.

ˆ Hence, we have by the deﬁnition of A(Fn) that ﬁn (δ) ˆ (δ) L(Pi) ω∃t ∈ Tδ t ≥ τη (ω) ∧ Fn(X (t)) ≈ 0 =0. 158 3 Standard Representation Theory

Therefore, we get ˆ ( (δ)( ))1 ( ) ≈ 0 ∈ ( ) Fn X ω,t (t≥τ (δ)(ω)) Pi dω ,si / A2 δ . Ω η

Hence, the approximation (3.2.15)holds.

Definition 3.2.5. (1) We say that a hyperfinite quadratic form E(·, ·) generates a δ-quasi-continuous semigroup if for all δ-quasi-continuous internal ∗ fin functions f : S0 −→ R, Ttf is δ-quasi-continuous for all t ∈ Tδ . (2) A hyperfinite quadratic form E(·, ·) generates a quasi-continuous semigroup if it generates a δ-quasi-continuous semigroup for some infinitesimal δ ∈ T.

Proposition 3.2.3. Let Y be a regular Hausdorff space, and let E(δ)(·, ·) be a hyperfinite weak coercive quadratic form for every infinitesimal δ∈ T . Assume that E(·, ·) has a hyperfinite δˆ-quasi-continuous core and is δˆ-nearstandardly concentrated for some δˆ ∈ T,δˆ ≈ 0. Moreover, suppose that there is an infinitesimal δ ∈ T such that E(·, ·) generates a δ-quasi-continuous semigroup for every infinitesimal δ ∈ T,δ ≥ δ . Then, there is an infinitesimal δ1 ∈ T and a δ1-exceptional set A1(δ1) such that (δ ) (i) For all si ∈ S0 −A1(δ1) and L(Pi)-a.e.ω, the path X 1 (ω,·) has S-right

and S-left limits at all t<ζδ1 (ω). (ii) For all inﬁnitely close si,sj ∈ S0 − A1(δ1), we have

◦ (δ1)+ ◦ (δ1)+ L(Pi){ X (ω,t) ∈ B} = L(Pj){ X (ω,t) ∈ B} (3.2.16)

for all ﬁnite t ∈ [0, ∞) and all Borel sets B. (iii) For all si ∈ S0 − A1(δ1), we have ( ) ∃ ∈ ﬁn ◦ ≥ ( ) ∧ ˆ ( (δ1)( )) ≈ 0 =0 L Pi ω t Tδ1 t ζδ1 ω Fn X t ,

ˆ ˆ for all Fn ∈ πˆ,whereπˆ = {Fn | n ∈ N}⊃π = {Fn | n ∈ N} is the countable set obtained in Lemma 3.2.5.

Proof. Step 1. Since E(·, ·) has a hyperﬁnite δˆ-quasi-continuous core, δˆ∈T,δˆ ≈ 0, there are a family of countable functions π = {Fn | n ∈ N}⊂H and a ˆ δ-exceptional set A such that Fn(si) ≈ Fn(sj ),si ≈ sj,si,sj∈S0 −A, ∀n ∈ N, and

◦ ◦ (Y − A ) ∩B(Y ) ⊂ σ{fn = F n | n ∈ N}. 3.2 Hyperﬁnite Dirichlet Forms and Markov Processes 159

Moreover, we have

st{ max |Fn(si)|} < ∞, si∈S0 ◦ (δˆ) E1 (Fn,Fn) < ∞,∀n ∈ N.

◦ (δˆ) By Lemma 3.2.7, π separates the points of Y − A by E1 (·, ·). From Propo- sition 3.2.1, there exists an inﬁnitesimal δ0 ∈ T such that for all inﬁnitesimal

δ ∈ Tδ0 ,thereisaδ-exceptional set A0(δ) such that for all si ∈ S0 − A0(δ), the hyperﬁnite Markov chain X(δ)(t) has S-left and S-right limits at all t<ζδ,L(Pi)-a.e.

Step 2. For simplicity, we may suppose δ0 ≥ δ and δ0 ∈ Tδˆ. Now let us ˆ take δ1 = δ0.Letπˆ = {Fn | n ∈ N}⊃π = {Fn | n ∈ N} be the countable set ∗ 1 m0 obtained in Lemma 3.2.5.Pickupm0 ∈ N − N such that 2 ≤ 2 δ1 ≤ 1. Set

m0 δ(1) = 2 δ1,δ(N +1) 1 = ( ) ≥ 2 ∈ N 2δ N ,N ,N .

Deﬁne for N ∈ N

N ˆ N Λ(N)= {TtFn | t ∈ Tδ(N)}. n=1

Then, {Λ(N) | N ∈ N} is an increasing sequence. N ˆ For each N ∈ N,t∈ Tδ(N),TtFn ∈ Λ(N) is hyperﬁnite δ1-quasi-continuous. Let us ﬁx N ∈ N. Then, there exists a sequence of internal sets {Bm,N | m ∈ N} decreasing as a function of m such that ˆ ˆ Fn(Xt(ω)) dPi(ω) ≈ Fn(Xt(ω)) dPj (ω), Ω Ω ∞ N for all si,sj ∈ S0 − m=1 Bm,N ,si ≈ sj,t∈ Tδ(N),n=1, 2, ··· ,N.Moreover, ∞ m=1 Bm,N is δ1-exceptional. Step 3. Set

( )= ( ) A1 δ1 A1 δ0 ∞ ∞ ˆ = A0(δ0) A2(δ) Bm,N , N=1 m=1 160 3 Standard Representation Theory ˆ ˆ where A2(δ) is obtained in Lemma 3.2.8 for δ = δ. Then, A1(δ1) is δ1- exceptional. Moreover, we have ˆ ˆ Fn(Xt(ω)) dPi(ω) ≈ Fn(Xt(ω)) dPj (ω), (3.2.17) Ω Ω ∞ N for si,sj ∈ S0 − A1(δ1),si ≈ sj,t ∈ N=1 Tδ(N),n ∈ N. Recalling the result obtained in step 1 and Lemma 3.2.8,wehavefrom(3.2.17)thatforall ˆ Fn ∈ π,ˆ t ∈ [0, ∞),si,sj ∈ S0 − A1(δ1),si ≈ sj, ◦ ˆ ◦ (δ1)+ ◦ ˆ ◦ (δ1)+ F n( X(t) ) dL(Pi)= F n( X (t)) dL(Pj). (3.2.18) Ω Ω

We deduce from the relation (3.2.18) and Lemma 3.2.6 that for all si,sj ∈ S0 − A1(δ1),si ≈ sj and t ∈ [0, ∞),theresult(3.2.16) holds. Hence, we have completed the proof of Proposition 3.2.3.

3.2.4 Construction of Strong Markov Processes

Let Y be a regular Hausdorff space, and let E(δ)(·, ·) be a hyperfinite weak coercive quadratic form for every infinitesimal δ ∈ T . Assume that E(·, ·) has a hyperfinite δˆ-quasi-continuous core and is δˆ-nearstandardly concentrated for some δˆ ∈ T,δˆ ≈ 0. Moreover, suppose that there is an infinitesimal δ ∈ T such that E(·, ·) generates a δ-quasi-continuous semigroup for every infinitesimal δ ∈ T,δ ≥ δ .Letδ1 ∈ T be the infinitesimal and let A1(δ1) be the δ1-exceptional set satisfying the results (i), (ii) and (iii) of Proposition 3.2.3.

Let x : Ω × R+ −→ YΔ be deﬁned by (δ) (i) if X (ω,0) ∈/ A1(δ1), then we have

(δ )+ x(ω,t)=◦X 1 (ω,t).

(δ) (ii) if X (ω,0) ∈ A1(δ1), then we have for all t ∈ R+

x(ω,t)=st(X(δ)(ω,0)).

◦ In the same way as in Sect. 3.1,wedenotebyA1(δ1) the inner standard part of A1(δ1), i.e., ◦ −1 A1(δ1) = y ∈ Y | st (y) ∩ S0 ⊂ A1(δ1) .

Lemma 3.2.9. For all inﬁnitely close si,sj ∈ S0 − A1(δ1), all ﬁnite sequences t1

n n ◦ ◦ (δ1)+ ◦ ◦ (δ1)+ ul( X (tl)) L(Pi)(dω)= ul( X (tl)) L(Pj)(dω) Ω l=1 Ω l=1 (3.2.19) and n n L(Pi) {x(tl) ∈ Bl} = L(Pj) {x(tl) ∈ Bl} . (3.2.20) l=1 l=1

Proof. The case of n =1is Proposition 3.2.3 (ii) and the relation (3.2.18). If =2 ˆ ˆ ∈ ﬁn ˆ ≈ ˆ ≈ ˆ ˆ n , let us pick t1, t2 Tδ1 such that t1 t1, t2 t2, t1 < t2 and

◦ (δ1) ˆ (δ0) ˆ u1(X (t1))u2(X (t2)) L(Pk)(dω) Ω

◦ ◦ (δ1)+ ◦ ◦ (δ1)+ = u1( X (t1)) u2( X (t2)) L(Pk)(dω),k = i, j. Ω

We have from the Markov property (δ1) ˆ (δ1) ˆ u1(X (t1))u2(X (t2)) Pk(dω) Ω

(δ1) (δ1) = ( (ˆ )) (δ ) ( (ˆ − ˆ )) ( ) = u1 X t1 EX 1 (tˆ1)u2 X t2 t1 Pk dω ,k i, j. Ω

Deﬁne a function f : Ω −→ R as follows:

◦ ◦ (δ1) ◦ ◦ (δ1) ( )= 1( ( 1)) 2( ( 2))1(t <ζ (ω)) f ω u X ω,t u X ω,t 2 δ1 .

Then, the function

(δ1) ˆ (δ1) ˆ ω −→ u1(X (t1))u2(X (t2)) is a lifting of f with respect to both Pi and Pj . Therefore, we have

◦ ◦ (δ1)+ ◦ ◦ (δ1)+ u1( X (t1)) u2( X (t2)) L(Pi)(dω) Ω (δ1) ˆ (δ1) ˆ ≈ u1(X (t1))u2(X (t2)) Pi(dω) Ω

(δ1) (δ1) = ( (ˆ )) (δ ) ( (ˆ − ˆ )) ( ) u1 X t1 EX 1 (tˆ1)u2 X t2 t1 Pi dω Ω 162 3 Standard Representation Theory ◦ ˆ ˆ ◦ ◦ (δ1) ˆ t1−t2 (δ1) ˆ ≈ u1( X (t1)) Q u2(X (t1)) L(Pi)(dω) Ω ◦ ˆ ˆ ◦ ◦ (δ1) ˆ t1−t2 (δ1) ˆ ≈ u1( X (t1)) Q u2(X (t1)) L(Pj)(dω) Ω

(δ1) (δ1) ≈ ( (ˆ )) (δ ) ( (ˆ − ˆ )) ( ) u1 X t1 EX 1 (tˆ1)u2 X t2 t1 Pj dω Ω (δ1) ˆ (δ1) ˆ = u1(X (t1))u2(X (t2)) Pj(dω) Ω

◦ ◦ (δ1)+ ◦ ◦ (δ1)+ ≈ u1( X (t1) u2( X (t2)) L(Pj)(dω). Ω

Nowitiseasytoshowthat(3.2.19)and(3.2.20)holdifn =2.Wecanthen complete the proof of Lemma 3.2.9 by mathematical induction.

In the following, we will deﬁne quantities {Πt} and {Θy} associated with ◦ our standard Markov process. For t ∈ R+,letΠt be the σ-algebra generated by the sets ωx(ω,t1) ∈ B1 ∧···∧x(ω,tn) ∈ Bn , where 0 ≤ t1 t

The ﬁltration {Πt | t ∈ [0, ∞)} is right continuous. Let us set

Π∞ = ∨{Πt | t ∈ [0, ∞)} .

◦ If y ∈ A1(δ1) ∪{Δ} ,C ∈ Π∞,let 1, if C contains all constant paths x(t)=y, Θy(C)= 0, else.

◦ If y/∈ A1(δ1) , let for all C ∈ Π∞

−1 Θy(C)=L(Pi)(C) for all si ∈ st (y) − A1(δ1).

Theorem 3.2.1. Let Y be a regular Hausdorff space, and let E(δ)(·, ·) be a hyperfinite weak coercive quadratic form for every infinitesimal δ ∈ T . Assume that E(·, ·) has a hyperfinite δˆ-quasi-continuous core and is δˆ-nearstandardly concentrated for some δˆ ∈ T,δˆ ≈ 0. Moreover, suppose that there is an infinitesimal δ ∈ T such that E(·, ·) generates a δ-quasi-continuous semigroup for every infinitesimal δ ∈ T,δ ≥ δ . Then, the associated 3.2 Hyperfinite Dirichlet Forms and Markov Processes 163 hyperfinite Markov chain has a modified standard part which is a strong Markov process.

Proof. The proof of this theorem follows exactly the same line as the proof of Theorem 3.1.1.

In Theorem 3.2.1, we assume that there is an inﬁnitesimal δ ∈ T such that E(·, ·) generates δ-quasi-continuous semigroup for every inﬁnitesimal δ ∈ T,δ ≥ δ . In practice, we have the following result:

Proposition 3.2.4. Assume that for each S-bounded u ∈D(E) there is a sequence {un | n ∈ N} of hyperﬁnite quasi-continuous functions such that ◦ ◦ |un − u|1 −→ 0 as n −→ ∞ . Then, all functions with E1(v, v) < ∞ are quasi-continuous, and the hyperﬁnite quadratic form E(·, ·) generates a quasi- continuous semigroup.

Remark 3.2.2. In Theorem 3.2.1, we obtain conditions on a hyperfinite weak coercive quadratic form to guarantee that the associated hyperfinite Markov chain has a modified standard part which is a strong Markov process, using the concepts of exceptional sets and semigroups. Correspondingly, one may obtain conditions on a hyperfinite weak coercive quadratic co-form to guarantee that the associated hyperfinite Markov chain has a modified standard part which is a strong Markov process, using the concepts of co-exceptional sets and co-semigroup. The details will not be presented here, since the construction is entirely similar to the one which leads to Theorem 3.2.1.

Chapter 4 Construction of Markov Processes Associated With Quasi-Regular Dirichlet Forms

In this chapter, we consider the construction of tight strong Markov processes associated with standard quasi-regular Dirichlet forms by using the language of nonstandard analysis. In Fan [165], the construction of strong Markov processes associated with standard symmetric Dirichlet forms was considered. In this chapter, we consider nonsymmetric Dirichlet forms. We consider a standard Dirichlet form (F (·, ·),D(F )) whose state space is a regular Hausdorﬀ space Y . We impose conditions directly on the Dirichlet form (F (·, ·),D(F )) to obtain associated tight strong Markov processes. The motivation comes from the desire to ﬁnd the relationship between the family of Dirichlet forms and the family of strong Markov processes.

Before describing our main result of Theorem 4.1.1, let us briefly summarize the development and applications of standard Dirichlet space theory. In the symmetric case, the regular Dirichlet space theory is well developed [175, 183, 333, 334]. For the nonsymmetric case, a similar theory is also developed [44, 105, 257]. The references mentioned above consider the case where the state space is a locally compact separable space (thus essentially a finite dimensional space, like Rd or a d-dimensional Riemannian manifold). This theory gives us, in particular, a very good understanding about the property of diffusion processes with non-smooth diffusion coefficients [45, 112, 161, 164, 240, 331, 346, 347].

In the last decades, an extensive development of inﬁnite dimensional stochastic analysis theory has taken place. In this case, the state space of the corresponding Dirichlet space theory is a general topological space, in particular not a locally compact separable space [2, 43, 50, 159, 160, 162, 163, 249, 295, 314–316, 326–328]. Applications in mathematical physics have been given [2, 48–50, 314].

In Albeverio et al. [25], a nonstandard symmetric Dirichlet space is proposed. In this book, we have generalized the corresponding results to nonsymmetric case in the Chaps. 1, 2,and3. Let us recall the main outline we have achieved in this book, which will be the basis for our probabilistic construction. In Sect. 1.5, we have discussed hyperﬁnite Dirichlet forms and

S. Albeverio et al., Hyperfinite Dirichlet Forms and Stochastic Processes, 165 Lecture Notes of the Unione Matematica Italiana 10, DOI 10.1007/978-3-642-19659-1_4, c Springer-Verlag Berlin Heidelberg 2011 166 4 Construction of Markov Processes hyperfinite Markov chains. In the Chap. 2, we have developed the potential theory of hyperfinite Dirichlet forms. In Sect. 3.1 of Chap. 3, we have studied the standard parts of hyperfinite Markov chains. Suppose that S0 is a hyperfinite subset of ∗Y for some topological space Y .IfX(t) is a hyperfinite Markov chain taking values in S = S0 ∪{s0}, we have got conditions that guarantee the existence of the standard part of X(t).Inaddition,we have showed that it is a Y -valued strong Markov process. After that, we have translated the conditions into the language of hyperfinite Dirichlet forms in Sect. 3.2.

The arrangement of this chapter is as follows. We will first present our main result, and related notations and definitions in Sect. 4.1. Then, we will construct hyperfinite quadratic forms of the standard Dirichlet forms in Sect. 4.2. In the meantime, we will construct hyperfinite Markov chains as well as dual hyperfinite Markov chains of the hyperfinite quadratic forms. The main aim of rest of the chapter is then devoted to show that the hyperfinite Markov chains have the desired projection of dual strong Markov processes from Sect. 4.3 to Sect. 4.6. In the last Sect. 4.7, we will prove the necessity part of our main result (Theorem 4.1.1).

4.1 Main Result

In this chapter, we assume that the state space Y is a regular Hausdorﬀ space or a T3 space, which is introduced in Sect. 3.2.LetB(Y ) be the topological Borel σ-ﬁeld of Y .Letν be a positive Radon measure on (Y,B(Y )) (see Sect. 4.2 below for details). Let supp(ν) be the smallest closed subset of Y such that ν(Y −supp(ν)) = 0. We suppose that Y = supp(ν) for convenience.

Deﬁnition 4.1.1. (1) A strong Markov process (Ω,Π,Πt,x(t),Θy) is called ν-tight if there exists an increasing sequence of compact sets {Yn | n ∈ N} of Y such that Θy lim σY −Y = ζ =1,ν-a.e.y ∈ Y, (4.1.1) n→∞ n

where σY −Yn is the hitting time of Y − Yn and ζ is the lifetime of x(t). That is,

σY −Yn =inf{t>0 | x(t) ∈ Y − Yn} (4.1.2) and

ζ(ω)=inf{t ≥ 0 | x(ω,t)=Δ} . 4.1 Main Result 167 ˆ (2) Two strong Markov processes (Ω,Π,Πt,x(t),Θy) and (Ω,Π,Πt,x(t), Θy) 2 are said to be ν-dual if for all t ∈ R+,u,v ∈ L (Y,ν), we have ˆ u(y)Θtv(y) ν(dy)= v(y)Θtu(y) ν(dy). Y Y

Let (F (·, ·),D(F )) be a coercive closed form on L2(Y,ν) as introduced 2 in Sect. 1.6 by replacing K with L (Y,ν). The inner product (·, ·)ν of K = L2(Y,ν) is deﬁned by 2 (f,f)ν = f (y)ν(dy),α ∈ [0, ∞). Y ˆ Let {Rα | α ∈ (−∞, 0)} and {Rα | α ∈ (−∞, 0)} be the resolvent and co-resolvent of (F (·, ·),D(F )), respectively. Let us denote by {Tt | t ∈ (0, ∞)} ˆ and {Tt | t ∈ (0, ∞)} the strong continuous contraction semigroups corre- ˆ sponding to {Rα | α ∈ (−∞, 0)} and {Rα | α ∈ (−∞, 0)}, respectively. Moreover, set

Fα(f,f)=F (f,f)+α(f,f)ν .

A bounded operator R on L2(Y,ν) is called a sub-Markovian operator if for every u ∈ L2(Y,ν), 0 ≤ u(x) ≤ 1,ν-a.e.x ∈ Y, we have 0 ≤ Ru(x) ≤ 1, ν-a.e.x ∈ Y. A coercive closed form F (·, ·) on L2(Y,ν) is called a Dirichlet form if it generates a semigroup {Tt | t ≥ 0} of sub-Markovian operators and ˆ a co-semigroup {Tt | t ≥ 0} of sub-Markovian operators.

Let (F (·, ·),D(F )) be a Dirichlet form on L2(Y,ν). For an open subset B of Y , let us deﬁne

L(B)={g ∈ D(F ) | g ≥ 1,ν-a.e. on B} . (4.1.3)

Suppose L(B) = ∅. From Ma and Röckner [270], Proposition 1.5 of Chap. III, there exists a unique γ1(B), γˆ1(B) ∈ L(B) such that for all w ∈ L(B)

F1(γ1(B),w) ≥ F1(γ1(B),γ1(B)), F1(w, γˆ1(B)) ≥ F1(ˆγ1(B), γˆ1(B)). (4.1.4)

From Ma and Röckner [270], Remark 1.6 and Exercise 1.7 of Chap. III, we know

F1(γ1(B),γ1(B)) = F1(γ1(B), γˆ1(B)) = F1(ˆγ1(B), γˆ1(B)). (4.1.5) 168 4 Construction of Markov Processes

Set

Γ1(B)=F1(γ1(B),γ1(B)) if L(B) = ∅ and Γ1(B)=∞ if L(B)=∅. (4.1.6)

For a general subset A of Y , let us deﬁne

Γ1(A)=inf{Γ1(B) | A ⊂ B,B is open} . (4.1.7)

We call Γ1(·) the 1-capacity of F (·, ·).

For a closed subset B ⊂ Y, we set

D(F )B = {f ∈ D(F ) | f =0,ν-a.e. on Y − B} .

It is easy to see that D(F )B is a closed set of D(F ) with the inner product F1(·, ·).

Deﬁnition 4.1.2. An increasing sequence of closed subsets {Gn | n ∈ N} of Y is called a nest if Γ1(Y − Gn) ↓ 0.

A subset B ⊂ Y is said to be of zero capacity if there exists a nest {Gn | ∞ n ∈ N} such that B ⊂ n=1(Y − Gn). A function f on Y is said to be quasi-continuous if there exists a nest {Gn | n ∈ N} such that the restriction f|Gn is continuous on Gn for each n ≥ 1.

We shall say that two ν-dual strong Markov processes (Ω,Π,Πt,x(t),Θy) ˆ and (Ω,Π,Πt,x(t), Θy) are properly associated with the Dirichlet form (F (·, ·),D(F )) if

ˆ ˆ 2 Ttf = Θtf and Ttf = Θtf,ν-a.e.,∀f ∈ L (Y,ν),t>0.

The main result of this chapter is the following:

Theorem 4.1.1. Assume that Y is a regular Hausdorﬀ topological space and ν is a Radon measure on Y .Let(F (·, ·),D(F )) be a Dirichlet form on L2(Y,ν). Then there are two ν-dual tight strong Markov processes ˆ (Ω,Π,Πt,x(t),Θy) and (Ω,Π,Πt,x(t), Θy) properly associated with the Dirichlet form (F (·, ·),D(F )) if and only if the following three conditions are satisﬁed:

(I) There is an increasing sequence of compact subsets {Yn | n ∈ N} of Y ( ) ( ) such that n∈N D F Yn is F1-dense in D F . (II) There exists a subset π0 of D(F ) such that π0 is dense in D(F ) with the inner product F1(·, ·). In addition, u is quasi-continuous for each u ∈ π0. 4.1 Main Result 169

(III) There is a countable subset π of D(F ) such that every element u ∈ π is quasi-continuous and bounded. Moreover, we have B(Y ) Yn ⊂ σ(u|u ∈ π), n∈N

where {Yn | n ∈ N} is a sequence of compact sets in (I).

Remark 4.1.1. If (F (·, ·),D(F )) satisﬁes conditions (I), (II) and (III) in Theorem 4.1.1,wecallitquasi-regular.

Remark 4.1.2. Previous work of Albeverio and Ma [41]andMaandRöckner [270] has established necessary and sufficient conditions for the existence of ν-perfect processes associated with Dirichlet forms. In these references, the state space Y is a metrizable space and ν is a σ-finite measure on Y. Thus, 2 there is a strictly positive function ψ ∈ L (Y,ν), 0 <ψ≤ 1. Set h = R1ψ, where {Rα | α>0} is the resolvent of F (·, ·). The authors defined the h- weighted capacity as in Röckner [313] and used it as a useful tool. In our case, Y is a regular Hausdorff space and m is a Radon measure. We do not have an h-weighted capacity, since this concept has not been developed in our context. Instead, we utilize the 1-capacity theory developed in Chap. 2, Fan [166], and Fukushima [175].

In order to prove Theorem 4.1.1, we will construct in Sect. 4.2 a hyperfinite quadratic form (E(·, ·), D(E)) from (F (·, ·),D(F )). We obtain a hyperfinite Markov chain X(t) associated with E(·, ·), as well as the dual Xˆ(t) of X(t). In Sect. 4.3, we will discuss the relation between the potential theory of (F (·, ·),D(F )) and the counterpart of its hyperfinite lifting (E(·, ·), D(E)). On this basis, we will consider the path regularity of X(t) and get its modified standard part x(t).Infact,x(t) is the wanted process. We shall show that x(t) satisfies all the requirements of Theorem 4.1.1 in Sect. 4.6. At last, we prove in Sect. 4.7 that if there are two ν-dual tight Markov ˆ processes (Ω,Π,Πt,x(t),Θy) and (Ω,Π,Πt,x(t), Θy) properly associated with F (·, ·), then x(t) satisfies the conditions (I), (II) and (III) of Theo- rem 4.1.1. Combining Propositions 4.6.1 and 4.7.1, we shall then prove the Theorem 4.1.1.

Remark 4.1.3. From Albeverio and Ma [41], we know that even if ν is a nowhere Radon measure on a metric space, it is still possible to get the Markov process x(t). In order to include this case by methods of nonstandard analysis, we would have to develop first a nonstandard measure theory associated with nowhere Radon measures on general topological spaces. After that, we could study the related stochastic analysis problems associated with a nonstandard version of nowhere Radon measure spaces. This is, however, a program which remains to be fulfilled. 170 4 Construction of Markov Processes 4.2 Hyperfinite Lifts of Quasi-Regular Dirichlet Forms

Let Y be a Hausdorﬀ topological space. Consider the topological Borel σ-ﬁeld B(Y ) on Y . Assume that ν is a positive Borel measure on (Y,B(Y )) such that ν(K) < ∞ for every compact set K.ItisaRadon measure in the sense that for all B ∈B(Y ),

ν(B)=sup{ν(K) | K ⊂ B,K compact} (4.2.1) and for all B ∈B(Y ) with ν(B) < ∞,

ν(B)=inf{ν(O) | B ⊂ O, O open} .

We denote by (Y,B(Y ),ν) the completion of (Y,B(Y ),ν). For simplicity, we suppose that supp(ν)=Y.

Let ∗Y be the nonstandard extension of Y . Given an element a ∈ Y ,we know that the monad of a is the subset of ∗Y deﬁned by μ(a)= {∗O | a ∈ O and O ⊂ Y is open} .

Apointy ∈ ∗Y is nearstandard if it belongs to μ(a) for some a ∈ Y. We shall say that it is nearstandard to a. The set of all nearstandard points is denoted by Ns(∗Y ). Since Y is Hausdorﬀ, each element y ∈ Ns(∗Y ) is nearstandard to exactly one element a ∈ Y (refer to page 48, 2.1.6. Proposition, [25]). We call a the standard part of y and denote it by st(y) or ◦y.

∗ ∗ Let Δ be an element outside Y , YΔ = Y ∪{Δ}. For all y ∈ Y − Ns( Y ), we deﬁne st(y)=Δ. Nowwehavegotthestandard part operation as a map

∗ st : Y −→ YΔ.

∗ ∗ A subset S0 of Y is called rich if it is hyperfinite and st(S0 ∩ Ns( Y )) = Y. ∗ In the following, we will construct a rich subset S0 in Y and an internal −1 measure m on S0 such that ν = L(m) ◦ st . By a finite B(Y ) partition of Y , we mean a finite collection Bi ∈ B(Y )1 ≤ i ≤ n

n of non-empty sets with Y = i=1 Bi and Bi ∩ Bj = ∅ if i = j. Let us denote by P the collection of all ﬁnite B(Y ) partitions of Y .

If P1 and P2 are elements of P, we say that P2 is ﬁner than P1 and we write P1 ≤ P2 if for each C ∈ P1,C = {B ∈ P2 | B ⊂ C}. 4.2 Hyperﬁnite Lifts of Quasi-Regular Dirichlet Forms 171

We have the following result from 1.1 Theorem, Loeb [266].

∗ ∗ Proposition 4.2.1. There is a partition P ∈ P such that P0 ≤ P for any ∗ P0 ∈P. That is, P ∈ P has the following properties: (i) There are an infinite integer N ∈ ∗N and an internal bijection from ∗ I = {i ∈ N | 1 ≤ i ≤ N} onto P . Thus we may write P = {Ai | i ∈ I}. (ii) If i and j are in I and i = j,thenAi = ∅,Aj = ∅ and Ai ∩ Aj = ∅. ∗ (iii) Y = {Ai | i ∈ I}. ∗ (iv) For each B∈ B(Y ), let IB = {i ∈ I | Ai ⊂ B}. Then IB is hyperfinite, ∗ = i and B i∈IB A . ∗ Hereafter, let us fix a partition P = {Ai | i ∈ I} of Y with above properties (i) through (iv). For every i ∈ I, we pick up an element si ∈ Ai and define S0 = {si | i ∈ I}. Let m be the internal measure on S0 defined by

∗ m({si})= ν(Ai).

∗ −1 Proposition 4.2.2. S0 is rich in Y and ν = L(m) ◦ st .

Proof. We simply refer to Albeverio et al. [25], 3.4.10 Corollary.

Let H be the hyperﬁnite dimensional space of all the internal functions ∗ f : S0 −→ R with the inner product f,g = fg dm. S0

Let us set K = L2(Y,ν). Let H˜ be the subspace of ∗K consisting of all ˜ functions which are constant on each class Aj,j ∈ I. Obviously, H and H are isomorphic. From now on, we will identify H with H˜ .ByusingTheorem1.6.1, we can prove the following result.

Proposition 4.2.3. Let Y be a Hausdorﬀ space, ν be a Radon measure on Y ,andF (·, ·) be a coercive closed form on L2(Y,ν). Then, we have ∗ (i) There exist a hyperﬁnite, rich subset S0 of Y , and an internal measure ∗ m on S0 such that H is S-dense in K. 2 (ii) There are a nonnegative quadratic form (E(·, ·), D(E)) on L (S0,m) and an internal time-line T = {kΔt | k ∈ ∗N} representing F (·, ·) in the following sense: ν = L(m) ◦ st−1 and for all u ∈ L2(Y,ν),

F (u, u)=inf{◦E(v, v) | v is a 2-lifting of u} .

Moreover, let {Qs | s ∈ T } and {Qˆs | s ∈ T } be the semigroup and co-semigroup generated by E(·, ·), respectively. Then for all t ∈ [0, ∞), ◦ s ∈ T,u ∈ K, v ∈ H such that t = s, u = stK (v), we have 172 4 Construction of Markov Processes

s ˆs ˆ stK Q v = Ttu and stK Q v = Ttu.

∗ ˆ ∗ On the other hand, let {Gα | α ∈ ( −∞, 0)} and {Gα | α ∈ ( −∞, 0)} be the resolvent and co-resolvent generated by E(·, ·), respectively. Then for all β ∈ (−∞, 0),α ∈ ∗( −∞, 0),u ∈ K, v ∈ H such that β = ◦α, u = stK (v), we have ˆ ˆ stK Gαv = Rβu and stK Gαv = Rβu,

ˆ where {Rα | α ∈ (−∞, 0)} is the resolvent, and {Rα | α ∈ (−∞, 0)} is the co-resolvent of (F (·, ·),D(F )).

Proof. From Sect. 3.2 of Albeverio et al. [25], we know that each function f in L2(Y,ν) has an S-square integrable lifting f˜ in H˜ such that ◦f˜(x)=f(◦x) for almost all nearstandard x. If we can show that ||∗f − f˜|| ≈ 0,thenH˜ is dense in ∗(L2(Y,ν)).By(4.2.1), it suﬃces to show this when f is bounded and of compact support. For such functions, the statement is an immediate consequence of Anderson’s nonstandard version of Lusin’s theorem. The proposition follows by Theorem 1.6.1.

Corollary 4.2.1. If the form F (·, ·) in Proposition 4.2.3 is a Dirichlet form, we can also take E(·, ·) to be a hyperﬁnite quadratic form.

Proof. We observe that from Proposition 1.5.1 it suﬃces to prove that QΔt and QˆΔt are Markov operators, where

QΔtu, v = u, QˆΔtv and for all u, v ∈ H 1 E(u, v)= (I − QΔt)u, v Δt 1 = u, (I − QˆΔt)v. Δt

∗ ∗ ∗ Let P be the projection of K onto H. We shall write { Rα | α ∈ (−∞, 0)} ∗ ∗ ∗ ˆ ∗ ˆ for ({Rα | α ∈ (−∞, 0)}) and { Rα | α ∈ (−∞, 0)} for ({Rα | α ∈ (−∞, 0)}), respectively. We see from Sect. 1.6 that QΔt and QˆΔt are just ∗ ∗ ˆ 1 −P (γRγ) and −P (γRγ) for a suitable inﬁnitesimal Δt = − γ , respectively. With the choice of H made in the proof of Proposition 4.2.3, this projection is just the conditional expectation with respect to the algebra generated by the partition P . Since conditional expectations preserve nonnegativity and decrease the supremum norm, the corollary follows.

In the rest of this chapter, we shall be interested in quadratic forms generating Markov processes. That is, we shall assume that F (·, ·) is a Dirichlet 4.2 Hyperﬁnite Lifts of Quasi-Regular Dirichlet Forms 173 form on L2(Y,ν). We shall use the results obtained above. First, let us introduce some notations.

The inﬁnitesimal generator A and co-generator Aˆ of E(·, ·) are given by

1 1 1 1 A = (I − QΔt) and Aˆ = (I − QˆΔt), 0 <Δt≤ = . Δt Δt ||A|| ||Aˆ||

The semigroup Qt and co-semigroup Qˆt,t∈ T, are deﬁned by

Qt =(QΔt)k and Qˆt =(QˆΔt)k for each t = kΔt.

Moreover, we have the following relation from Proposition 1.6.3, F (u, u) = lim −α u(y) u(y)+αRαu(y) ν(dy) α−→ − ∞ Y ◦ ∗ ∗ = lim [−α(I + αRα)P u,P u] α→−∞ α≈−∞ α∈∗(−∞,0) ◦ ∗ ∗ = lim [−α(I + αGα)P u,P u] α→−∞ α≈−∞ α∈∗(−∞,0) ◦ = lim [(α)E(u, u)] (4.2.2) α→−∞ α≈−∞ α∈∗(−∞,0) for every u ∈ D(F ),whereP is the projection of ∗K onto H. Therefore, we see from the proof of Theorem 1.6.1 that 1 (I − QΔt)P ∗u, P ∗u≈F (u, u), Δt i.e.,

◦E(P ∗u,P ∗u)=F (u, u) < ∞ for each u ∈ D(F ). (4.2.3)

We recall from Definition 1.4.2 that the domain D(E) of E(·, ·) is the set of all u ∈ H such that ◦ ◦ (i) E1(u, u)= [E(u, u)+u, u] < ∞. ∗ (ii) For all infinite α ∈ (−∞, 0), E(u + αGαu, u + αGαu) ≈ 0 and E(u + ˆ ˆ αGαu, u + αGαu). We remark that E(·, ·) does not necessarily satisfy the hyperfinite weak sector condition. Hence, the results of Proposition 1.4.2 need not hold. How- ever, it is easy to see from the proof of Proposition 1.4.2 the following: if ◦E(u, u) < ∞ and for all infinite α<0, (α)E(u, u) ≈E(u, u) and (α)Eˆ(u, u) ≈E(u, u),thenwehaveu ∈D(E). 174 4 Construction of Markov Processes

Therefore, we have proved the following using (4.2.2)and(4.2.3):

Lemma 4.2.1. For every u ∈ D(F ), we have P ∗u ∈D(E).

Now, we are in the position to construct the dual processes X(t) and Xˆ(t) and their transition matrices associated with E(·, ·). Remember that m is an internal measure on the hyperﬁnite set S0 = {s1,s2, ··· ,sN }. We shall write mi for m(si) if it is convenient. ∗ Fix j ∈ I. Let uj : S0 −→ R be the function satisfying uj(sl)=δlj,l∈ I. We deﬁne

Δt ˆ ˆΔt bij = Q uj(si) and bij = Q uj(si),j ∈ I.

∗ It is easy to see that for any internal function u : S0 −→ R, we have

N u(·)= u(sj)uj(·). j=1

Hence, we obtain

N Δt Δt Q u(si)= u(sj)Q uj(si) j=1 N = bij u(sj), j=1 ⎛ ⎞ N 1 ⎝ ⎠ Au(si)= u(si) − u(sj)bij . Δt j=1

Similarly, we have

N ˆΔt ˆΔt Q u(si)= u(sj)Q uj(si) j=1 N ˆ = bij u(sj), j=1 ⎛ ⎞ N ˆ 1 ⎝ ˆ ⎠ Au(si)= u(si) − u(sj)bij . Δt j=1

Let us denote u(si) by u(i). Then, we have 4.2 Hyperﬁnite Lifts of Quasi-Regular Dirichlet Forms 175

1 N E(u, v)= Au(i)v(i)m(i) Δt i=1 ⎡ ⎤ N N 1 ⎣ ⎦ = u(i) − bij u(j) v(i)m(i) Δt i=1 j=1 1 N N = (δij − bij )u(j)v(i)m(i) Δt i=1 j=1 N N 1 ˆ = (δij − bij )v(j)u(i)m(i). Δt i=1 j=1

Since the duality of QΔt and QˆΔt,wehaveforalli, j ∈ I, ˆ m(i)bij = m(j)bji.

Recall that s0 is a point outside S0, and we put S = S0 ∪{s0}. We assume m({s0})=0and st(s0)=Δ. Let Q = {qij | 0 ≤ i, j ≤ N} be an (N +1)× (N +1)matrix satisfying the following conditions:

q00 =1, q0i =0 for i ∈ I, qij = bij ,i,j ∈ I, N qi0 =1− bij ,i∈ I. j=1

It is easy to see that

N qij =1 for all i ∈ I ∪{0} . j=0

We construct a hyperﬁnite Markov chain {X(t) | t ∈ T } associated with Q and m in the following manner. Let Ω be the set of all internal functions ω : T −→ S. Let X be the coordinate function X(ω,t)=ω(t).WetakeP to be the measure generated by

P ([ω]0)=m(X(ω,0)) and k−1 P ([ω]kΔt)=m {X(ω,0)} q(ω(nΔt),ω((n +1)Δt)), n=0 176 4 Construction of Markov Processes where q(si,sj) is just qij and [ω]t is deﬁned by

[ω]t = {ω | X(ω ,s)=X(ω,s) for all s ≤ t} .

Let Ft be the internal algebra on Ω generated by the sets [ω]t,ω ∈ Ω. Let us deﬁne the probability measure Pi as follows:

k−1 Pi([ω]kΔt)=δiω(0) q(ω(nΔt),ω((n +1)Δt)). n=0

It is easy to verify that the following Markov property of X(t) holds. If X(ω,t)=si, then we have P ω ∈ [ω]tX(ω ,t+ Δt)=sj = qij P ([ω]t).

qˆ00 =1, qˆ0i =0 for i ∈ I, ˆ qˆij = bij ,i,j ∈ I, N ˆ qˆi0 =1− bij ,i∈ I. j=1

It is easy to see that

N qˆij =1 for all i ∈ I ∪{0} . j=0

We may construct the dual hyperﬁnite Markov chain {Xˆ(t)=X(t) | t ∈ T } ˆ ˆ and the corresponding internal measure P,Pi,i∈ I.

If δ ∈ T ,letTδ be the sub-line

∗ Tδ = {kδ | k ∈ N0} .

(δ) (δ) We write X for the restriction X|Tδ. For each t ∈ Tδ,letFt be the internal algebra on Ω generated by the sets (δ) (δ) (δ) [ω]t = ω ∈ ΩX (ω ,s)=X (ω,s) for all s ∈ Tδ,s≤ t . 4.3 Relation with Capacities 177

(δ) (δ) (δ) In the following of this chapter, we shall use the notations A , qij , E (·, ·), and D(E(δ)) introduced in Sect. 1.5.

For δ ∈ T,α ∈ ∗R,α≥ 0, let us introduce the notation

(δ) (δ) Eα (u, u)=E (u, u)+αu, u.

4.3 Relation with Capacities

Let (F (·, ·),D(F )) be a Dirichlet form on L2(Y,ν). We consider the 1-capacity Γ1(·) deﬁned in the relations (4.1.3), (4.1.4), (4.1.5), (4.1.6), and (4.1.7)in Sect. 4.1. We get the following results from Ma and Röckner [270], Chap. III.

Lemma 4.3.1. Let B be an open subset of Y such that L(B) = ∅,where L(B) is deﬁned by the deﬁnition (4.1.3)inSect.4.1. Then we have the following results:

(1) There exists a unique element γ1(B) ∈ L(B) such that

Γ1(B)=F1(γ1(B),γ1(B)). (4.3.1)

(2) 0 ≤ γ1(B) ≤ 1, ν-a.e. and γ1(B)=1, ν-a.e. on B.

Proposition 4.3.1. Γ1(·) has the properties:

(1) A ⊂ B =⇒ Γ1(A) ≤ Γ1(B). (2) An ↑=⇒ Γ1(∪n∈NAn)=supn∈N Γ1(An). (3) A, B open =⇒ Γ1(A ∪ B)+Γ1(A ∩ B) ≤ Γ1(A)+Γ1(B). (4) Γ1(∪n∈NAn) ≤ n∈N Γ1(An).

In the same way as in Sects. 2.2 and 2.3, we introduce the capacity theory associated with (E(δ)(·, ·), D(E(δ))) and state some properties without proof.

∗ For δ ∈ T,α ∈ R+, and an internal subset A of S0, we deﬁne

(δ) σA (ω)=min {t ∈ Tδ | X(ω,t) ∈ A} , (δ) (δ) −σA /δ eα (A)(i)=Ei (1 + αδ) , ∗ L(A)={G | S0 −→ R is internal and G|A =1A} .

(δ) Using the terminology of Sects. 2.2 and 2.3,wecalleα (A) the equilibrium α-potential of A associated with E(δ)(·, ·). Let us set (δ) (δ) (δ) (δ) Capα (A)=Eα eα (A),eα (A) . 178 4 Construction of Markov Processes

For an arbitrary subset B of S0, we deﬁne (δ) (δ) Capα (B)=inf Capα (A)B ⊂ A, A is internal .

(δ) (δ) We call Capα (·) the α-capacity associated with E (·, ·). We have studied the potential theory of hyperﬁnite quadratic forms in Chap. 2. We shall use the same notations of Chap. 2 in the following.

We ﬁrst remark that the weak sector condition |F1(x, y)|≤C F1(x, x) F1(y,y) for all x, y ∈ D(F ) is equivalent to the following: |F (x, y)|≤C F1(x, x) F1(y,y) for all x, y ∈ D(F ), for some nonnegative constant C ∈ [0, ∞). Obviously, we can always suppose that C = C. Hence, we have |F (x, y)|≤C F1(x, x) F1(y,y) for all x, y ∈ D(F ). (4.3.2)

Proposition 4.3.2. Let C satisfy the inequality (4.3.2). For all open subset B ⊂ Y, we have for all δ ∈ T,δ ≈ 0 the following

(δ) ∗ 2 Cap1 ( B ∩ S0) ≤ (C +1) Γ1(B).

Proof. For simplicity, we may suppose that L(B) = ∅. From Lemma 4.3.1, there is an element γ1(B) ∈ D(F ) satisfying (4.3.1). For convenience, let us assume that γ1(B)(x)=1for all x ∈ B. For all δ ∈ T,δ ≈ 0,wehavefrom Proposition 2.2.1

(δ) ∗ (δ) (δ) (δ) Cap1 ( B ∩ S0)=E1 (e1 (1∗B),e1 (1∗B)) (δ) (δ) ∗ = E1 (e1 ( B),γ1(B)). (4.3.3)

On the other hand, let γ be given by (1.6.5) in the proof of Theorem 1.6.1, Chap. 1. Then, we have

(δ) (δ) ∗ ∗ (δ) ∗ E (e1 ( B),γ1(B)) = −γ(I + (γRγ))e1 ( B),γ1(B) (γ) (δ) ∗ = F (e1 ( B),γ1(B)). (4.3.4)

From the inequality (4.3.2) and Corollary 1.6.1 (i), we get 4.3 Relation with Capacities 179

(γ) (δ) ∗ (δ) ∗ F (e1 ( B),γ1(B)) = F ((−γRγ )e1 ( B),γ1(B)) (δ) ∗ (δ) ∗ ≤ C F1((−γRγ)e1 ( B), (−γRγ)e1 ( B)) F1(γ1(B),γ1(B)). (4.3.5)

From Corollary 1.6.1 (ii), we have

(δ) ∗ (δ) ∗ F1((−γRγ)e1 ( B), (−γRγ)e1 ( B)) (δ) ∗ (δ) ∗ (δ) ∗ (δ) ∗ = F ((−γRγ)e1 ( B), (−γRγ)e1 ( B)) + (−γRγ )e1 ( B), (−γRγ)e1 ( B) (γ) (δ) ∗ (δ) ∗ (δ) ∗ (δ) ∗ ≤ F (e1 ( B),e1 ( B)) + e1 ( B),e1 ( B) ∗ (δ) ∗ (δ) ∗ (δ) ∗ (δ) ∗ = −γ(I + (γRγ ))e1 ( B),e1 ( B) + e1 ( B),e1 ( B) (δ) (δ) ∗ (δ) ∗ = E1 (e1 ( B),e1 ( B)). (4.3.6)

Therefore, we have from the relations (4.3.3), (4.3.4), (4.3.5), and (4.3.6)that

◦ ◦ (δ) ∗ (δ) (δ) (δ) Cap1 ( B ∩ S0) = E1 (e1 (1∗B),e1 (1∗B)) 2 ≤ (C +1) F1(γ1(B),γ1(B)).

This ﬁnishes the proof of Proposition 4.3.2.

Corollary 4.3.1. Let A ⊂ Y satisfy the condition Γ1(A)=0. Then for all (δ) ∗ δ ∈ T,δ ≈ 0, we have Cap1 ( A ∩ S0) ≈ 0.

Proof. The proof follows easily from Proposition 4.3.2.

Corollary 4.3.2. Let {Gn | n ∈ N} be a sequence of increasing closed subsets of Y satisfying Γ1(Y − Gn) ↓ 0. Then for all δ ∈ T,δ ≈ 0, we have ◦ ∞ (δ) ∗ Cap1 S0 − ( Gn ∩ S0) =0. (4.3.7) n=1

Proof. From Proposition 4.3.2,wehave

◦ ◦ (δ) ∗ (δ) ∗ Cap1 (S0 − Gn ∩ S0) = Cap1 ( (Y − Gn) ∩ S0) 2 ≤ (C +1) Γ1(Y − Gn) −→ 0,n→∞.

Therefore, (4.3.7)holds.

Proposition 4.3.3. Let {Gn | n ∈ N} be a sequence of increasing closed ∞ ∗ subsets of Y satisfying Γ1(Y − Gn) ↓ 0. Then S0 − n=1 Gn ∩ S0 is δ- exceptional for all δ ∈ T,δ ≈ 0.

Proof. The proof follows easily from Corollary 4.3.2 and Theorem 2.4.1. 180 4 Construction of Markov Processes 4.4 Path Regularity of Hyperﬁnite Markov Chains

Deﬁnition 4.4.1. Let Z be a subset of Y . We say that a subset π of L2(Y,ν) separates points of Z, if for any two diﬀerent points x and y in Z,thereexists u ∈ π such that u(x) = u(y).

Lemma 4.4.1. If the hypothesis (III) in Theorem 4.1.1 holds, then π separates the points of n∈N Yn.

Proof. The proof can be carried through in the same way as the proof of Lemma 3.2.7.

Lemma 4.4.2. Assume that Y is a regular Hausdorﬀ space and the hypothesis (III) of Theorem 4.1.1 holds. Let δ ∈ T,δ ≈ 0. For every ω ∈ Ω, if the path X(δ)(ω,t) fails to have an S-left or S-right limit at some t<τ(δ)(ω), then so does (P ∗u)(X(δ)(ω,t)) for some u ∈ π, where the stopping time τ (δ)(ω) is deﬁned by (δ) ◦ (δ) ∗ τ (ω)=inf tX (ω,t) ∈/ ( Yn ∩ S0) . n∈N

∗ Proof. Let us set Bn = Yn ∩ S0,n ∈ N and B = ∪n∈NBn. For each n ∈ N, we deﬁne (δ) (δ) τn (ω)=min t ∈ T X (ω,t) ∈/ Bn , ◦ (δ) ζδ(ω)=inf tX (ω,t) ∈/ S0 ,

(δ) i.e., ζδ is the lifetime of X (ω,t). It is easy to show that

(δ)( ) ≤ ( ) τ ω ζδ ω, (δ) ◦ (δ) τ (ω)=sup τn (ω)n ∈ N .

Let ω be a ﬁxed point in Ω.Fixt<τ(δ)(ω),t ∈ [0, ∞). Then, there is an (δ) element N ∈ N such that t<τN (ω). Given a sequence {tn | n ∈ N} from T ◦ such that the standard part tn increases strictly to t, we can show that the sequence ◦ (δ) X (ω,tn)n ∈ N has a cluster point in YN in same way as the proof of Lemma 3.2.1.Therest follows in a similar way. 4.5 Quasi-Continuity and Nearstandard Concentration 181

Proposition 4.4.1. Assume that Y is a regular Hausdorﬀ space and the hypothesis (III) of Theorem 4.1.1 holds. Let δ ∈ T,δ ≈ 0. Then there exists a δ-exceptional set A0(δ) such that for all si ∈ S0 − A0(δ), the hyperﬁnite (δ) Markov chain X (t) has S-left and S-right limits at all t<ζδ,L(Pi)-a.e.

Proof. The result can be proved by using the method of Proposition 3.2.1, and the results of Lemmas 4.2.1 and 4.4.2.

4.5 Quasi-Continuity and Nearstandard Concentration

Let (F (·, ·),D(F )) be a Dirichlet form on L2(Y,ν). First of all, we present some properties of E(·, ·).

Lemma 4.5.1. Let {fk | k ∈ N} be a sequence of quasi-continuous functions on Y . Then there exists a nest {Gn | n ∈ N} such that for all k ∈ N, fk ∈ C({Gn}),where

C({Gn})={f | f|Gn is continuous for each n ∈ N} .

Proof. The proof is similar to the one in Fukushima [175] Theorem 3.1.2.

Lemma 4.5.2. (1) If the condition (I) in Theorem 4.1.1 is satisﬁed, we have

F1(u − un,u− un) −→ 0 as n −→ ∞ , ∀u ∈ D(F ),

where un ∈ D(F )Yn is the projection of u.

(2) Suppose that the condition (II) in Theorem 4.1.1 holds. Then, every element f ∈ D(F ) admits a quasi-continuous version.

Proof. (1) Let F (·, ·) be the symmetric part of F (·, ·), i.e., 1 ( )= ( )+ ( ) ∈ ( ) F u, v 2 F u, v F v, u ,u,v D F .

Set

F 1(u, v)=F (u, v)+(u, v),u,v ∈ D(F ).

For every u ∈ D(F ), we have the following decomposition with respect to the inner product F 1(·, ·)

u =(u − un)+un,un ∈ D(F )Yn ,un − u⊥D(F )Yn , ∀n ∈ N.

Let vn = u − un. Then, we have 182 4 Construction of Markov Processes

F 1(vn,vn+m)=F 1(vn+m,vn+m)+F 1(vn − vn+m,vn+m) = F 1(vn+m,vn+m)+F 1(un+m − un,u− un+m) = F 1(vn+m,vn+m)+F 1(−un,u− un+m).

Since un ∈ D(F )Yn ⊂ D(F )Yn+m , F 1(−un,u− un+m)=0. Hence, we have

F 1(vn,vn+m)=F 1(vn+m,vn+m).

Therefore, we have

F1(vn − vn+m,vn − vn+m)=F1(vn,vn) − F1(vn+m,vn+m) −→ 0,n→∞.

This means that {vn | n ∈ N} is a F1(·, ·) Cauchy sequence. Hence, there is an element v∞ ∈ D(F ) such that

F1(vn − v∞,vn − v∞) −→ 0,n−→ ∞ .

It is easy to see that

F1(v∞,v)=0, ∀v ∈ D(F )Yn . n∈N ( ) ( ) =0 Since n∈N D F Yn is F1-dense in D F , we know that v∞ . This proves (1).

(2) For this proof, we refer to Albeverio and Ma [41], 4.1 Lemma and Fukushima [175], Theorem 3.1.3. Lemma 4.5.3. Suppose that the condition (I) in Theorem 4.1.1 is satisﬁed. Then (F1(·, ·),D(F )) is a separable Hilbert space.

Proof. Since Yn is compact, (F1(·, ·),D(F )Yn ) is separable. Hence (F (·, ·), D(F )) is separable. Lemma 4.5.4. Suppose that the hypotheses (I), (II) and (III) in Theo- rem 4.1.1 hold. Then, there exists a countable set πˆ such that

(1) π ⊂ πˆ ⊂ D(F ) ∩ Qb(Y ), (4.5.1) (2) u, v ∈ π,ˆ a ∈ Q =⇒|u|∈π,ˆ u + v ∈ π,ˆ uv ∈ π,ˆ av ∈ πˆ and u ∧ a ∈ π,ˆ (4.5.2) (3)π ˆ is F1-dense in D(F ), where Q = {a1,a2, ··· ,an, ···} is the set of all rational numbers and Qb(Y ) is the space of all bounded quasi-continuous functions deﬁned on Y. Moreover, there is a nest {Zn | n ∈ N} such that for every u ∈ πˆ, u ∈ C({Zn}). 4.5 Quasi-Continuity and Nearstandard Concentration 183

Proof. From Lemmas 4.5.2 and 4.5.3, we can assume that π is F1-dense in D(F ). Deﬁne the maps

Λ = {λ−2,λ−1,λ0,λ1,λ2, ···} from [D(F ) ∩ Qb(Y )] × [D(F ) × Qb(Y )] into D(F ) ∩ (Qb(Y ) by

λ−2(u, v)=|u|, λ−1(u, v)=u + v, λ0(u, v)=uv, λ2i−1(u, v)=aiu, λ2i(u, v)=u ∧ ai,i=1, 2, ···

Applying Lemma 3.2.4 to π and Λ, we get a countable set πˆ satisfying the relations (4.5.1)and(4.5.2). Obviously, πˆ is F1-dense in D(F ). The existence of {Zn | n ∈ N} follows from Lemma 4.5.1.

Lemma 4.5.5. Assume that the hypotheses (I), (II) and (III) in Theo- rem 4.1.1 hold. For any two ﬁnite measures ν1 and ν2 on (Y,B(Y )) satisfying the following conditions:

u(y) ν1(dy)= u(y) ν2(dy) for all u ∈ π,ˆ Y Y where πˆ is the countable set obtained in the Lemma 4.5.4, we have that ν1 and ν2 coincide on (Y,σ(u|u ∈ πˆ)).

Proof. The proof is similar to the one of Lemma 3.2.6.

We recall that we have introduced the hyperﬁnite quasi-continuity and related concepts of (E(·, ·), D(E)) in Sect. 3.2.Wethenhave

Proposition 4.5.1. Let us assume that f : Y −→ R is a quasi-continuous function. Then for all δ ∈ T,δ ≈ 0,P∗f is hyperﬁnite δ-quasi-continuous.

Proof. The proof follows easily from Corollary 4.3.2 and Theorem 2.4.1.

Corollary 4.5.1. Let us assume that the condition (II) in Theorem 4.1.1 holds. Then for every element f ∈ D(F ) and all inﬁnitesimal δ ∈ T ,there ∗ ∗ is a δ-exceptional set A(δ) ⊂ S0 such that (P f)(si) ≈ (P f)(sj ) for all inﬁnitely close si,sj ∈ S0 − A(δ).

Proof. From Lemma 4.5.2 (2), there is a quasi-continuous version f˜ of f. Hence, P ∗f˜ is hyperﬁnite δ-quasi-continuous, ∀δ ∈ T,δ ≈ 0. Since P ∗f ≈ P ∗f˜, we prove Corollary 4.5.1 by using Lemma 2.1.1 (i). 184 4 Construction of Markov Processes

Proposition 4.5.2. Let Y be a regular Hausdorff space. If the conditions (I), (II), and (III) in Theorem 4.1.1 are satisfied, then there exist a δ0-exceptional ( ) ( ) ∈ 1 ≤ 1 set A1 δ0 of X t for some infinitesimal δ0 and some β Tδ0 , 2 <β such that (δ ) (i) For all si ∈ S0 − A1(δ0),thepathX 0 (ω,t) has S-right and S-left ( ) ( ) limits at all t<ζδ0 ω,L Pi -a.e.ω. ˆ ∞ (ii) Let us set Q(δ, β)= n=1 T2−nβ. For all infinitely close si,sj ∈ S0 − ˆ A1(δ0),s∈ Q(δ0,β),u∈ πˆ, we have

◦ ◦ ∗ ∗ (δ0) (P T st(s)u)(si) = (P u)(X (s)) Pi(dω) Ω ◦ ∗ (δ0) = (P u)(X (s)) Pj(dω). (4.5.3) Ω

◦ ∗ (iii) For all si ∈ S0 −A1(δ0),u∈ π,ˆ [P (T tu)(si)] : [0, ∞) −→ R is continu- ∈ − ( ) ∈ ˆ ∈ ﬁn ous with respect to t. Moreover, for all si S0 A1 δ0 ,u π, s Tδ0 , we have

∗ s ∗ P (T st(s)u)(si) ≈ Q (P u)(si).

Proof. Step 1. For s ∈ T ﬁn, let t =st(s). If u ∈ π,ˆ P ∗u is a 2-lifting of u. s ∗ Therefore, Q (P u) is a 2-lifting of Ttu from Proposition 4.2.3. For all k ∈ N, we have s ∗ ∗ 1 1 μ si ∈ S0|Q (P u)(si) − P (Ttu)(si)| > < . (4.5.4) k k

Therefore, (4.5.4) holds for some inﬁnite k ∈ ∗N − N also. This means that s ∗ ∗ siQ (P u)(si) ≈ P (Ttu)(si) is contained in an internal set of inﬁnitesimal measure. Having this in mind, we shall show our proposition step by step.

∗ ∗ Step 2. Suppose that πˆ = {un | n ∈ N}. Let {P un | n ∈ N} be an ∗ 1 ∗ internal extension of {P un | n ∈ N}. For ε ∈ T ,N ∈ N, deﬁne ﬁn s ∗ ∗ B1(ε)= si∃s ∈ Tε Q (P un)(si) ≈ P (Tst(s)un)(si) , n∈N N N s ∗ ∗ B1(ε, N)= si∃s ∈ Tε Q (P un)(si) ≈ P (Tst(s)un)(si) , (4.5.5) n=1 N ˆ N s ∗ ∗ 1 B1(ε, N)= si∃s ∈ Tε |Q (P un)(si) − P (Tst(s)un)(si)| > . n=1 N 4.5 Quasi-Continuity and Nearstandard Concentration 185

Fix ε ≈ 0,ε∈ T 1. For each N ∈ N, we have

ˆ 1 μ(B1(ε, N)) < . N Consider the following internal set 1 ˆ 1 1 ε ∈ T μ(B1(ε, N(ε)) < ,N(ε)= +1. N(ε) ε

1 By saturation, there is an inﬁnitesimal ε1 ∈ T such that ˆ 1 μ B1(ε1,N(ε1)) < . N(ε1)

In view of our Lemma 2.1.1, there is an inﬁnitesimal δ1 ∈ Tε such that ˆ 1ˆ B1(ε1,N(ε1)) is δ-exceptional for all δ ≥ δ1,δ ∈ Tε1 . Therefore, B1(δ, N(δ)) is δ-exceptional for all δ ∈ Tε1 ,δ ≥ δ1. Noticing that ˆ B1(δ) ⊂ B1(δ, N(δ)) ⊂ B1(δ, N(δ)),

we know that B1(δ) is δ-exceptional for all δ ∈ Tε1 ,δ ≥ δ1.

∈ ∗N − N 1 2n0 ≤ 1 =2n0 Step 3. We take n0 satisfying 2 < δ1 . Let β δ1 and ˆ ∞ Q(β)= n=1 T2−nβ. Deﬁne ˆ s ∗ s ∗ B2 = si ∈ S0∃s ∈ Q(β) Q (P u)(si) ≈ Q (P u)(sj) for some sj ≈ si . u∈πˆ

∗ Since P (Tst(s)u) is hyperﬁnite δ-quasi-continuous for all δ ∈ T1,δ ≈ 0, s ∗ ∗ and Q (P u) ≈ P (Tst(s)u), it follows from Lemma 2.1.1 that there is an inﬁnitesimal δ2 ∈ T such that B2 is δ-exceptional for all δ ≥ δ2,δ ∈ T.

n −m Step 4. We pick an infinite number m0 ≤ n0 such that δ0=2ˆ 0 0 δ1 ≈ 0 and δ0 ≥ δ2.LetA0(δ0) be the δ0-exceptional set obtained in Proposi- tion 4.4.1. Then, there exists a δ0-exceptional set A1(δ0) which contains A0(δ0),B1(δ0) and B2. Obviously, A1(δ0) satisfies (i). For si ∈ S0 − ( ) ∈ ˆ ∈ fin A1 δ0 ,u π,s Tδ0 , we have from the relations (4.5.5)that

s ∗ ∗ Q (P u)(si) ≈ P (T st(s)u)(si),

◦ ∗ since Tδ0 ⊂ Tδ1 ⊂ Tε1 . Thus, (P (Ttu)(si)) : [0, ∞) −→ R is continuous. This completes the proof of Proposition 4.5.2. 186 4 Construction of Markov Processes

Lemma 4.5.6. Suppose that the condition (I) in Theorem 4.1.1 holds. For every u ∈ D(F ),s∈ [0, ∞),theset ∗ ∗ B(u, s)= si ∈ S0 − ( Yn ∩ S0)P (Tsu)(si) ≈ 0 n∈N is Δt-exceptional. Proof. From Lemma 4.5.2,wehave

F1(u − un,u− un) −→ 0 as n −→ ∞ , ∀u ∈ D(F ),

where un ∈ D(F )Yn is the projection of u. For each k ∈ N,letustakenk ∈ N such that = ( − ( ) − ( ) ) 2−k δk F1 Tsu Tsu nk ,Tsu Tsu nk < ,

where (Tsu)n ∈ D(F )Yn is the projection of Tsu. For simplicity, we may ( s )n |Y −Y =0 suppose that T u k nk . Deﬁne ( )= ∈ − ∗ ∩ | ∗( )( )|≥ ∈ N Bk u, s si S0 Ynk S0 P Tsu si δk ,k .

Noticing that ∗ ∗ | ( s )( i) − ( s ) ( i)|≥ k i ∈ k( ) P T u s P T u nk s δ ,s B u, s , we have P ω∃t ≤ 1(X(ω,t) ∈ Bk(u, s)) ≤ ∃ ≤ 1 | ∗( )( ( )) − ∗( ) ( ( ))|≥ P ω t P Tsu X ω,t P Tsu nk X ω,t δk ≤ ∃ ≤ 1 ∗( )( ( )) − ∗( ) ( ( )) ≥ P ω t P Tsu X ω,t P Tsu nk X ω,t δk + ∃ ≤ 1 ∗( )( ( )) − ∗( ) ( ( )) ≤− P ω t P Tsu X ω,t P Tsu nk X ω,t δk . (4.5.6)

Let us set ∗ ∗ = { ∈ 0 | ( s )( i) − ( s ) ( i) ≥ k} A i S P T u s P T u nk s δ , ( )=min{ ∈ | ( ) ∈ } σA ω t T X t A , −σA/Δt e1(A)(i)=Ei (1 + Δ) . 4.5 Quasi-Continuity and Nearstandard Concentration 187

Then, we have ∃ ≤ 1 ∗( )( ( )) − ∗( ) ( ( )) ≥ P ω t P Tsu X ω,t P Tsu nk X ω,t δk

= P (ω|σA(ω) ≤ 1) −σA/Δt −1/Δt = Pi ω (1 + Δt) ≥ (1 + Δt) dm(i) S 0 −σA/Δt 1/Δt ≤ Ei (1 + Δt) (1 + Δt) dm(i) S0 1/Δt =(1+Δt) e1(A)(i) dm(i) S0 1/Δt ≤ (1 + Δt) E1(e1(A),e1(A)), (4.5.7) where the reason for the last step holding is (2.3.1) in Sect. 2.3.

From Theorem 2.2.1,wehave 1 E ( ( ) ( )) ≤E ( ) √ ( ∗( ) − ∗( ) ) 1 e1 A ,e1 A 1 e1 A , P Tsu P Tsu nk δk 1 = (γ) ( ) √ ( ∗( ) − ∗( ) ) F e1 A , P Tsu P Tsu nk δk 1 +√ ( )( )( ∗( ) − ∗( ) )( ) ( ) e1 A i P Tsu P Tsu nk i dm i δk S 0 1 = (− ) ( ) √ ( ∗( ) − ∗( ) ) F γRγ e1 A , P Tsu P Tsu nk δk 1 +√ ( )( )( ∗( ) − ∗( ) )( ) ( ) e1 A i P Tsu P Tsu nk i dm i , δk S0 (4.5.8) where γ is given by (1.6.5) in the proof of Theorem 1.6.1. From the inequality (4.3.2) in Sect. 4.3,wehave 1 (− ) ( ) √ ( ∗( ) − ∗( ) ) F γRγ e1 A , P Tsu P Tsu nk δk 1 ≤ C √ F1 ((−γRγ )e1(A), (−γRγ )e1(A)) δk × ( ∗( ) − ∗( ) ∗( ) − ∗( ) ) F1 P Tsu P Tsu nk ,P Tsu P Tsu nk . (4.5.9)

In the same way as for the relation (4.3.6) in Sect. 4.3, we can show that

F1 ((−γRγ)e1(A), (−γRγ )e1(A)) ≤E1(e1(A),e1(A)). 188 4 Construction of Markov Processes

Hence, we have from the inequality (4.5.9)that 1 (− ) ( ) √ ( ∗( ) − ∗( ) ) F γRγ e1 A , P Tsu P Tsu nk δk 1 ≤ C √ E1(e1(A),e1(A)) δk × ( ∗( ) − ∗( ) ∗( ) − ∗( ) ) F1 P Tsu P Tsu nk ,P Tsu P Tsu nk . (4.5.10)

From the relations (4.5.8)and(4.5.10), we have 1 E1(e1(A),e1(A)) ≤ (C +1)√ E1(e1(A),e1(A)) δk × ( ∗( ) − ∗( ) ∗( ) − ∗( ) ) F1 P Tsu P Tsu nk ,P Tsu P Tsu nk .

Therefore, we have

2 (C +1) ∗ ∗ ∗ ∗ E1(e1(A),e1(A)) ≤ F1 P (Tsu) − P (Tsu)nk ,P (Tsu) − P (Tsu)nk δk 2 ≤ (C +1) δk. (4.5.11)

From the relations (4.5.7)and(4.5.11), we have ∃ ≤ 1 ∗( )( ( )) − ∗( ) ( ( )) ≥ P ω t P Tsu X t P Tsu nk X t δk

1/Δt 2 ≤ (1 + Δt) (C +1) δk. (4.5.12)

1/Δt 2 ≤ (1 + Δt) (C +1) δk. (4.5.13)

From the relations (4.5.6), (4.5.12), and (4.5.13), we have 1/Δt 2 P ω∃t ≤ 1 X(ω,t) ∈ Bk(u, s) ≤ 2 1+Δt (C +1) δk

1/Δt ≤ 2 1+Δt (C +1)22−k. (4.5.14) 4.5 Quasi-Continuity and Nearstandard Concentration 189

∞ For each K ∈ N, we have B(u, s) ⊂ k=K Bk(u, s). It follows from the inequality (4.5.14)thatB(u, s) is Δt-exceptional.

Lemma 4.5.7. Suppose that the condition (I) in Theorem 4.1.1 is satis- ﬁed. Then, there exists a Δt-exceptional set A2 containing all B(u, s),u∈ π,ˆ s ∈ Q+, where Q+ is the family of all positive rational numbers.

Proof. This follows easily from Lemma 2.1.1 and Lemma 4.5.6.

(δ) In Sect. 4.4, we have deﬁned the lifetime ζδ of X (ω,t).Thatis,forδ ∈ T , we have ◦ (δ) ζδ(ω)=inf t | X (ω,t) ∈/ S0 .

◦ (δ)+ (δ) Now, we deﬁne the right standard part X of X as follows. If t<ζδ(ω), let

◦X(δ)+(ω,t)=S- lim X(δ)(ω,s) s↓t

(δ) whenever this limit exists and for all t1

Proposition 4.5.3. Assume that Y is a regular Hausdorff space. If the conditions (I), (II), and (III) in Theorem 4.1.1 are satisfied, then there exists a properly δ0-exceptional set A(δ0) of X(t) for some infinitesimal δ0 and some ∈ 1 ≤ 1 β Tδ0 , 2 <β such that

(δ ) (i) For all si ∈ S0 −A(δ0), the path X 0 (ω,t) has S-right and S-left limits

at all t<ζδ0 (ω),L(Pi)-a.e.ω. ∈ − ( ) ∈ ˆ ∈ ﬁn (ii) For all inﬁnitely close si,sj S0 A δ0 ,u π, t Tδ0 , we have

◦ (δ0) ◦ (δ0) ( ( )) ( i)( )= ( ( ))1(t<ζ ) ( i)( ) u X ω,t L P dω u X ω,t δ0 L P dω Ω Ω ◦ (δ0) = ( ( ))1(t<ζ ) ( j)( ) u X ω,t δ0 L P dω Ω

◦ (δ0) = u( X (ω,t)) L(Pj)(dω). (4.5.15) Ω

(iii) For all inﬁnitely close si,sj ∈ S0 −A(δ0), all ﬁnite t ∈ [0, ∞), and every Borel set B ∈B(Y ), we have ◦ (δ0)+ ◦ (δ0)+ L(Pi) X (ω,t) ∈ B = L(Pj) X (ω,t) ∈ B . (4.5.16) 190 4 Construction of Markov Processes

◦ ∗ (iv) For all si ∈ S0 − A(δ0),u∈ π,ˆ [P (T tu)(si)] : [0, ∞) −→ R is continu- ∈ − ( ) ∈ ˆ ∈ ﬁn ous with respect to t. Moreover, for all si S0 A δ0 ,u π,s Tδ0 , we have ∗ s ∗ P (T st(s)u)(si) ≈ Q (P u)(si).

Proof. Let A2 be the Δt-exceptional set obtained in Lemma 4.5.7 and let A1(δ0) be the δ0-exceptional set obtained in Proposition 4.5.2, δ0 ∈ T,δ0 ≈ 0. Let A(δ0) be a properly δ0-exceptional set containing A1(δ0), A2,and ∗ S0 − n∈N( Zn ∩S0),where{Zn | n ∈ N} is the nest obtained in Lemma 4.5.4.

Let {Yn | n ∈ N} be the increasing sequence of compact sets in (I). Let ∗ ∗ us set Bn = Yn ∩ S0,n ∈ N,andB = ∪n∈NBn. Let {Bn | n ∈ N} be an increasing extension of {Bn | n ∈ N}. Deﬁne (δ0)( )= ∈ (δ0)( ) ∈ ∈ ∗N τn ω min t Tδ0 X ω,t / Bn ,n , (δ0) ◦ (δ0) τ (ω)=inf tX (ω,t) ∈/ B .

We have (δ0 ) ◦ (δ0) τ (ω)=sup τn (ω)n ∈ N . (4.5.17)

∗ Given si,sj ∈ S0 − A(δ0),si ≈ sj, we can ﬁnd an η ∈ N − N such that

(δ0 ) ◦ (δ0) τ (ω)= τη (ω),L(Pi) and L(Pj)-a.e.

(δ0) Since Yn is compact, n ∈ N, we have from (4.5.17)thatτ (ω) ≤ ζδ0 (ω), ∀ω ∈ Ω. Let us pick an internal stopping time σ(ω) such that

◦ (δ0) L(Pk)( σ = ζδ0 )=1,k = i, j, and τη (ω) ≤ σ(ω), ∀ω ∈ Ω.

∈ ﬁn ∈ ˆ If t Tδ0 ,u π, we have

∗ (δ0) ∗ (δ0) (P u)(X (t)) Pi(dω)= (P u)(X (t))1 (δ0) Pi(dω) (t<τη ) Ω Ω + ( ∗ )( (δ0)( ))1 ( ) P u X t (t≥τ (δ0)) Pi dω . Ω η (4.5.18)

From Proposition 4.5.2 (iii) and Lemma 4.5.7,wehave ( ∗ )( (δ0)( ))1 ( ) ≈ 0 = P u X t (t≥τ (δ0)) Pk dω ,k i, j. (4.5.19) Ω η 4.5 Quasi-Continuity and Nearstandard Concentration 191

∈ ﬁn It follows from the relations (4.5.3), (4.5.18), and (4.5.19) that for all t Tδ0

◦ ◦ ( ∗ )( (δ0)( ))1 ( )= ( ∗ )( (δ0)( )) ( ) P u X t (t<τ (δ0)) Pi dω P u X t Pi dω Ω η Ω ◦ ∗ (δ0) = (P u)(X (t)) Pj(dω) Ω ◦ = ( ∗ )( (δ0)( ))1 ( ) P u X t (t<τ (δ0)) Pj dω . Ω η (4.5.20)

Therefore, we have from the relations (4.5.19)and(4.5.20) that for all t ∈ ˆ Q(δ0,β)

◦ ◦ (δ0) ∗ (δ0) ( ( ))1(t<ζ ) ( i)( )= ( )( ( ))1(t<σ) i( ) u X t δ0 L P dω P u X t P dω Ω Ω ◦ = ( ∗ )( (δ0)( ))1 ( ) P u X t (t<τ (δ0)) Pi dω Ω η ◦ ∗ (δ0) = (P u)(X (t)) Pi(dω) Ω ◦ ∗ (δ0) = (P u)(X (t)) Pj(dω) Ω ◦ = ( ∗ )( (δ0)( ))1 ( ) P u X t (t<τ (δ0)) Pj dω Ω η ◦ ∗ ◦ (δ0) = (P u)( X (t))1(t<σ) Pj(dω) Ω ◦ (δ0) = ( ( ))1(t<ζ ) ( j)( ) u X t δ0 L P dω . Ω (4.5.21)

∈ ﬁn From Proposition 4.5.2 (iii), we see that (4.5.15)holdsforallt Tδ0 .For all inﬁnitely close si,sj ∈ S0 − A(δ0),t∈ [0, ∞),u∈ π,ˆ we have from (4.5.21) that ◦ (δ0)+ ◦ (δ0)+ u( X (t)) L(Pi)(dω)= u( X (t)) L(Pj)(dω). (4.5.22) Ω Ω ◦ (δ0)+ ∞ Because X (ω,t) ∈ n=1 Yn {Δ},L(Pk)-a.e.ω,k = i, j, we get (4.5.16) from (4.5.22) and Lemma 4.5.5. This completes the proof of Proposition 4.5.3. 192 4 Construction of Markov Processes 4.6 Construction of Strong Markov Processes

In this section, we suppose that the conditions (I), (II) and, (III) in Theo- rem 4.1.1 are satisﬁed. Let A(δ0) be the properly δ0-exceptional set of X(t) in Proposition 4.5.3, δ0 ≈ 0. Let us set (δ0 ) ◦ (δ0) ∗ τ (ω)=inf tX (ω,t) ∈/ ( Yn ∩ S0) , n∈N

where {Yn | n ∈ N} is the sequence of compact sets in (I) of Theorem 4.1.1. Let x : Ω × R+ −→ YΔ be deﬁned by

◦ (δ0)+ (δ ) (i) if X(ω,0) ∈/ A(δ0), then x(ω,t)= X (ω,t) if t<τ 0 (ω) and x(ω,t)=Δ if t ≥ τ (δ0)(ω). (ii) if X(ω,0) ∈ A(δ0), then x(ω,t)=st(X(ω,0)) for all t ∈ R+.

We call x the modiﬁed standard part of X(ω,t).

◦ In the same way as in Sect. 3.1,wedenotebyA(δ0) the inner standard part of A(δ0), i.e., ◦ −1 A(δ0) = y ∈ Y | st (y) ∩ S0 ⊂ A(δ0) .

Lemma 4.6.1. For all inﬁnitely close si,sj ∈ S0 −A(δ0), all ﬁnite sequences t1

n n ◦ (δ0)+ ◦ (δ0)+ ul( X (tl)) L(Pi)(dω)= ul( X (tl)) L(Pj)(dω) Ω l=1 Ω l=1 (4.6.1)

and n n L(Pi) {x(tl) ∈ Bl} = L(Pj) {x(tl) ∈ Bl} . (4.6.2) l=1 l=1

Proof. The case of n =1is part of Proposition 4.5.3.Ifn =2, let us pick ˆ ˆ ∈ ﬁn ˆ ≈ ˆ ≈ ˆ ˆ t1, t2 Tδ0 such that t1 t1, t2 t2, t1 < t2,and

◦ ∗ (δ0) ˆ ∗ (δ0) ˆ (P u1)(X (t1))(P u2)(X (t2)) L(Pk)(dω) Ω

◦ (δ0)+ ◦ (δ0)+ = u1( X (t1))u2( X (t2)) L(Pk)(dω),k = i, j. Ω 4.6 Construction of Strong Markov Processes 193

We have from the Markov property ∗ (δ0) ˆ ∗ (δ0) ˆ (P u1)(X (t1))(P u2)(X (t2)) Pk(dω) Ω

∗ (δ0) ∗ (δ0) = ( )( (ˆ )) (δ ) ( )( (ˆ − ˆ )) ( ) = P u1 X t1 EX 0 (tˆ1) P u2 X t2 t1 Pk dω ,k i, j. Ω

Deﬁne a function f : Ω −→ R as follows:

◦ (δ0) ◦ (δ0) ( )= ( ( )) ( ( ))1 δ f ω u1 X ω,t1 u2 X ω,t2 (t2<τ 0 (ω)).

Then, the function

∗ (δ0) ˆ ∗ (δ0) ˆ ω → (P u1)(X (t1))P u2(X (t2)) is a lifting of f with respect to both Pi and Pj . Therefore, we have

◦ (δ0)+ ◦ (δ0)+ u1( X (t1))u2( X (t2)) L(Pi)(dω) Ω ∗ (δ0) ˆ ∗ (δ0) ˆ ≈ (P u1)(X (t1))(P u2)(X (t2)) Pi(dω) Ω

∗ (δ0) ∗ (δ0) = ( )( (ˆ )) (δ ) ( )( (ˆ − ˆ )) ( ) P u1 X t1 EX 0 (tˆ1) P u2 X t2 t1 Pi dω Ω ◦ ˆ ˆ ◦ (δ0) ˆ t1−t2 ∗ (δ0) ˆ ≈ u1( X (t1)) Q (P u2)(X (t1)) L(Pi)(dω) Ω

◦ (δ0) ◦ (δ0) = u1( X (t1))(Tt2−t1 u2)( X (t1)) L(Pi)(dω) Ω

◦ (δ0) ◦ (δ0) = u1( X (t1))(Tt2−t1 u2)( X (t1)) L(Pj)(dω) Ω ◦ ˆ ˆ ◦ (δ0) ˆ t2−t1 ∗ (δ0) ˆ ≈ u1( X (t1)) Q (P u2)(X (t1)) L(Pj)(dω) Ω

∗ (δ0) ∗ (δ0) ≈ ( )( (ˆ )) (δ ) ( )( (ˆ − ˆ )) ( ) P u1 X t1 EX 0 (tˆ1) P u2 X t2 t1 Pj dω Ω ∗ (δ0) ˆ ∗ (δ0) ˆ = (P u1)(X (t1))(P u2)(X (t2)) Pj(dω) Ω

◦ (δ0)+ ◦ (δ0)+ ≈ u1( X (t1)u2( X (t2)) L(Pj)(dω). Ω

Now, it is easy to show that (4.6.1)and(4.6.2)holdifn =2.Wecancomplete the proof of Lemma 4.6.1 by mathematical induction. ˆ In the following, we will deﬁne quantities {Πt}, {Θy},and{Θy} associated ◦ with our standard Markov processes. For t ∈ R+,letΠt be the σ-algebra 194 4 Construction of Markov Processes generated by the sets ωx(ω,t1) ∈ B1 ∧···∧x(ω,tn) ∈ Bn , where 0 ≤ t1 t

The ﬁltration {Πt | t ∈ [0, ∞)} is right continuous. Let us set

Π∞ = ∨{Πt | t ∈ [0, ∞)} .

◦ If y ∈ A(δ0) ∪{Δ},C ∈ Π∞,let 1, if C contains all constant paths x(t)=y, Θy(C)= 0, else.

◦ If y/∈ A(δ0) , let for all C ∈ Π∞

−1 Θy(C)=L(Pi)(C) for all si ∈ st (y) − A(δ0).

Given a Radon probability measure μ on Y, we deﬁne two new measures μ0 and μ1 on Y by

◦ μ0(B)=μ(B − A(δ0) ), ◦ μ1(B)=μ(B ∩ A(δ0) ).

By Corollary 3.1.1, we can ﬁnd an internal measure λ on S0 satisfying

−1 μ0 = L(λ) ◦ st , L(λ)(A(δ0)) = 0. (4.6.3) 4.6 Construction of Strong Markov Processes 195

We first prove that (Ω,Π,Πt,x(t),Θy) satisfies Definition 3.1.1 (i):

(i) Given a Borel set E ⊂ Y, and t ∈ R+, we must show that

y → Θy(x(t) ∈ E) is μ-measurable for all ﬁnite Radon measures μ.Sinceforα ∈ [0, 1] ◦ ◦ A(δ0) ∩ E, α ∈ [0, 1) {y ∈ A(δ0) | Θy {x(t) ∈ E} >α} = ∅,α=1 is universally measurable by Lemma 3.1.2, it suﬃces to show that

◦ Y − A(δ0) y → Θy(x(t) ∈ E) is μ0-measurable. Fix u ∈ πˆ.From(4.5.20), (4.5.21), and (4.5.22) in Sect. 4.5,wecanchoose tˆ≈ t so large that

◦ ∗ (δ0) ˆ ◦ (δ0)+ (P u)(X (t)) Pλ(dω)= u( X (t)) L(Pλ)(dω). Ω Ω

Deﬁne a function f : Y −→ R as follows: ◦ (1) If y/∈ A(δ0) , let us deﬁne

◦ (δ0)+ f(y)= u( X (t)) L(Pi)(dω) Ω = Θtu(y)

−1 for some si ∈ st (y) ∩ S0 − A(δ0); ◦ (2) If y ∈ A(δ0) , deﬁne f(y) arbitrarily. The function (δ0) ˆ si → u(X (t))Pi(dω) Ω is a lifting of f with respect to λ. Therefore, y → Θtu(y) is μ0-measurable. For a ∈ Q, we can show that the map

y → Θy {u(x(t)) >a} is μ0-measurable, since un =[n(u − u ∧ a)] ∧ 1 ∈ πˆ and un ↑ 1(u>a). Hence,

y → Θy {x(t) ∈ E} is μ0-measurable for all E ∈B(Y ) by a monotone class Theorem. 196 4 Construction of Markov Processes ˆ Similarly, we may prove that (Ω,Π,Πt,x(t), Θy) also satisﬁes Deﬁnition 3.1.1 (i).

(v) In order to show that (Ω,Π,Πt,x(t),Θy) satisﬁes (3.1.9) in Sect. 3.1,it suﬃces to prove that for all {Πt} stopping times σ, u1,u2 ∈ πˆ and s ∈ [0, ∞), we have

EΘy (u1(x(σ))u2(x(σ + s))) μ(dy) Y = ( ( )) ( ( )) ( ) EΘy u1 x σ EΘx(σ) u2 x s μ dy . (4.6.4) Y

First we notice that since the paths of x are constant Θμ1 -a.e., we have

EΘy (u1(x(σ))u2(x(σ + s))) μ1(dy) Y = ( ( )) ( ( )) ( ) EΘy u1 x σ EΘx(σ) u2 x s μ1 dy . Y

Equation (4.6.4) will hold if we can prove

EΘy (u1(x(σ))u2(x(σ + s))) μ0(dy) Y = ( ( )) ( ( )) ( ) EΘy u1 x σ EΘx(σ) u2 x s μ0 dy . (4.6.5) Y

In the sequel, we will show that (4.6.5)holds.Letν be the nonstandard representation of μ0 given in the construction (4.6.3). We pick an internal ◦ = ( ) ˆ ∈ ﬁn ˆ ≥ ˆ ≈ stopping time τ such that τ σ, L Pν -a.e and s Tδ0 , s s, s s such that

u1(x(σ))u2(x(σ + s)) L(Pν )(dω) Ω

∗ (δ0) ∗ (δ0) ≈ (P u1)(X (τ))(P u2)(X (τ +ˆs)) Pν (dω). Ω

Then, we have

EΘy (u1(x(σ))u2(x(σ + s))) μ0(dy) Y

= u1(x(σ))u2(x(σ + s)) Θμ0 (dω) Ω

= u1(x(σ))u2(x(σ + s)) L(Pi)(dω) L(λ)(dsi) S0 Ω 4.7 Necessity for Existence of Dual Tight Markov Processes 197

= u1(x(σ))u2(x(σ + s)) L(Pλ)(dω) Ω ◦ ∗ (δ) ∗ (δ) = (P u1)(X (τ))(P u2)(X (τ +ˆs)) Pλ(dω) Ω ◦ ∗ (δ) ∗ (δ) = (P u1)(X (τ))EX(δ) (τ)(P u2)(X (ˆs)) Pλ(dω) Ω = ( ( )) ( ( )) ( )( ) u1 x σ EΘx(σ) u2 s L Pλ dω Ω = ( ( )) ( ( )) ( ) u1 x σ EΘx(σ) u2 s Θμ0 dω Ω = ( ( )) ( ( )) ( ) EΘy u1 x σ EΘx(σ) u2 s μ0 dy Y ˆ This is (4.6.5). Similarly, we can also prove that (Ω,Π,Πt,x(t), Θy) satis- ﬁes (3.1.9) in Sect. 3.1.

4.7 Necessity for Existence of Dual Tight Markov Processes

2 Lemma 4.7.1. Let (F (·, ·),D(F)) be a Dirichlet form on L (Y,ν). If the condition (I) holds, then ν(Y − n∈N Yn)=0. ( ) ( ) Proof. Since n∈N D F Yn is F1-dense in D F ,it is a dense subset of 2( ) =0 − ∀ ∈ 2( ) L Y,ν . This implies that f ,ν-a.e. on Y n∈N Yn, f L Y,ν . Since ν is a Radon measure, we deduce that ν(Y − n∈N Yn)=0. Proposition 4.7.1. Let (F (·, ·),D(F )) be a Dirichlet form on L2(Y,ν).If there exist ν-tight dual strong Markov processes (Ω,Π, Πt,x(t),Θy) and ˆ (Ω,Π,Πt,x(t), Θy) which are properly associated with F (·, ·), then conditions (I), (II) and (III) of Theorem 4.1.1 hold.

Proof. (I) Let {Rα | α ∈ (−∞, 0)} be the resolvent of x(t). Then, the family 2 {R1f | f ∈ L (Y,ν)} is a F1-dense subset of D(F ). Let {Yn | n ∈ N} be an increasing sequence of compact sets satisfying the condition (4.1.1)in

Sect. 4.1.DenotebyσY −Yn the hitting time of Y −Yn deﬁned by the deﬁnition (4.1.2) in Sect. 4.1.Forf ∈ L2(Y,ν),set σY −Yn n −t R1 f(y)=EΘy e f(xt) dt. 0 198 4 Construction of Markov Processes

n Noticing that R1 f ∈ D(F )Yn is the projection of R1f, we have

n n+m n n+m F1(R1 f − R1 f,R1 f − R1 f) n n n+m n+m = F1(R1 f,R1 f) − F1(R1 f,R1 f) −→ 0,n−→ ∞ .

Moreover, it follows from the condition (4.1.1) in Sect. 4.1 that

n R1 f −→ R1f,ν-a.e.

Hence, we have

n n F1(R1 f − R1f,R1 f − R1f) −→ 0,n→∞. ( ) ( ) Therefore, n∈N D F Yn is F1-dense in D F . 2 (II) For all f ∈ L (Y,ν),R1f is quasi-continuous by Theorem 4.3.3, 2 Fukushima [175]. Take π0 = {R1f | f ∈ L (Y,ν)}. (III) Let us set Z = n∈N Yn. Then, we know that the relative topology of Z is second countable. We pick a countable family of open sets such that {An ∩ Z | n ∈ N} forms a basis for the relative topology of Z and A ∩ Z is contained in a compact subset of Y .Foreachn ∈ N, we deﬁne σY −An −t un(y)=EΘy e 1An (xt) dt, 0

where σY −An is the hitting time of Y − An. From Lemma 4.4.2 of Fukushima [175], we have that un is quasi-continuous. Moreover, un satisﬁes

un(y) > 0 for y ∈ An, un(y)=0 for y ∈ Y − An.

This means that π ˆ={un | n ∈ N} satisﬁes (III) in Theorem 4.1.1.

Combining Propositions 4.6.1 and 4.7.1, we complete the proof of Theo- rem 4.1.1. Chapter 5 Hyperﬁnite Lévy Processes

So far, we have concentrated in this book on studying arbitrary Markov processes which admit a modiﬁcation whose paths are right-continuous with left limits (i.e. càdlàg Markov or Feller processes,cf.e.g.[307, Chap. 3, Theorem 2.7]), and their relations to energy forms and Hamiltonians – from the perspective of nonstandard analysis.

Independently from this theory, Tom Lindstrøm [263] developed a different approach of finding nonstandard representations for a special class of Feller processes and their infinitesimal generators, viz. stochastically continuous processes with stationary and independent increments, for short: Lévy processes.

In terms of the state space, we will have to narrow the scope of our investigation: The state space should now be at least a separable Hilbert space, and in this chapter, we will only treat Lévy processes on Rd.

Nevertheless, it is rather surprising to see how confining oneself to this, still fairly large, class of Markov processes reduces the amount of technicalities involved in developing a theory of internal hyperfinite representations of these processes, their semigroups and their infinitesimal generators.

Lindstrøm’s theory was carried further by Albeverio and Herzberg [14] who proved that the jump part of any Lévy process can be conceived of, in a natural and rigorous sense, as the hyperfinite sum of independent hyperfinite Poisson processes. In addition, Lindstrøm [264] introduced nonlinear stochastic integration with respect to (hyperfinite) Lévy processes, Herzberg [205] developed a theory of pathwise integration with respect to hyperfinite Lévy processes with bounded-variation jump part, and Herzberg and Lindstrøm [206] proved an internal jump-diffusion decomposition. Whilst there are also other approaches to Lévy processes from a nonstandard-analysis viewpoint, due to Albeverio and Herzberg [15]andNg[288], Lindstrøm’s theory of hyperfinite Lévy processes is the most promising for potential theory and other applications, in particular in equilibrium asset pricing (see [204]).

S. Albeverio et al., Hyperﬁnite Dirichlet Forms and Stochastic Processes, 199 Lecture Notes of the Unione Matematica Italiana 10, DOI 10.1007/978-3-642-19659-1_5, c Springer-Verlag Berlin Heidelberg 2011 200 5 Hyperﬁnite Lévy Processes and Applications

In this chapter, we will present the most important results in Lindstrøm’s original paper [263] and their extensions by Albeverio and Herzberg [14]. The thrust of the first sections of this Chapter is Lindstrøm’s work, though we shall provide much more detailed proofs and choose a slightly different approach that does not put as much emphasis on the jump-diffusion decomposition. En passant, we shall give detailed proofs of some of the results presented in Albeverio et al. [25], as well as outlining the link between hyperfinite Lévy processes (which are going to be defined in Sect. 5.2) and certain hyperfinite nonnegative quadratic forms.

Finally, we will brieﬂy discuss a few applications of the theory presented in this chapter.

5.1 Standard Lévy Processes

There is a vast literature on standard Lévy processes. A selection of references can be found in the monograph by Applebaum [65] and papers by Albeverio, Mandrekar, and Rüdiger [46], Marinelli, Prévôt and Röckner [277], Albeverio and Rüdiger [52] as well as by Albeverio, Rüdiger, and Wu [55]. First, we shall give the precise deﬁnition of a Lévy process. As previously, R+ will be short hand for the half-open interval [0, +∞). Deﬁnition 5.1.1. Consider a probability space (Ω,C,P) and let d ∈ N.A d stochastic process x : Ω × R+ → R is called a Lévy process if and only if

1. x0 =0P -almost surely, 2. xt − xs is independent of σ(xu | u ≤ s), 3. the law of xt − xs equals the law of xt−s,and

4. P -almost all paths of (xt)t∈R+ are right-continuous with left limits (càdlàg). Much of the appeal of the area of Lévy processes is due to representation results of the following kind (cf. e.g. [307, 323]): Theorem 5.1.1 (Lévy-Khintchine formula). Consider a probability d space (Ω,C,P) and let d ∈ N. A stochastic process x : Ω × R+ → R is a Lévy process if and only if there are a symmetric d × d-matrix with nonnegative entries C, a vector γ ∈ Rd, and a Radon measure ν on Rd \{0} satisfying | |2 ( ) +∞ 1. B1(0) y ν dy < , 2. ν[B1(0)] < +∞, such that for all y ∈ Rd and t ≥ 0, t ity · γ − 2 y · Cy+ iy·xt E e =exp ity·z . (5.1.1) d − 1 − · ( ) t R \{0} e iy zχB1(0) ν dz 5.1 Standard Lévy Processes 201

Here, Bρ(x) – as usual – is the open ball of radius ρ centered at x, for any d x ∈ R and ρ ∈ R>0. ( )={ ∈ ∗Rd || − | In internal contexts, however, we will also write√ Bρ x y y x ∗ ∗ d ∗ d <ρ} for ρ ∈ R>0 and x ∈ R (wherein |z| = z · z, for arbitrary z ∈ R , is the Euclidean norm of z,and· :(x, y) → x · y denotes the inner product on ∗Rd). Speaking rather informally, we suppress the “ ∗” when referring to ∗ the -image of the operator B·(·):(ρ, x) → Bρ(x).

A proof of this result is beyond the scope of this book, but it can be found, in the language of Fourier transforms, e.g. in the monographs by Bertoin [86] or Sato [323]. One can also formulate an analogous result for inﬁnitesimal generators of Lévy processes – a proof of the equivalence of the formulations by means of Fourier transforms and inﬁnitesimal generators can be found in the volume by Revuz and Yor [307].

In order to give a more handy formulation of Theorem 5.1.1 and later results, we will introduce the following terminology.

Deﬁnition 5.1.2. If x· =(xt)t≥0 is the Lévy process associated with a continuous and translation-invariant Markovian semigroup p· =(pt)t≥0,then the measure ν, the vector γ and the matrix C of (5.1.1) will also be referred to as the Lévy measure of x, the drift vector of x and the covariance matrix of x, respectively.

Deﬁnition 5.1.3. A Radon measure ν on Rd \{0} satisfying | |2 ( ) +∞ 1. B1(0) y ν dy < , 2. ν[B1(0)] < +∞ (equivalently, (1 ∧|y|2) ν(dy) < +∞), is called a Lévy measure. A triple d d×d (γ,C,ν) with γ ∈ R , C ∈ R≥0 symmetric, and ν a Lévy measure, will be referred to as a generating triplet.

There is a one-to-one correspondence between continuous translation- invariant Markovian semigroups and Lévy processes in the following sense.

Consider a probability space (Ω,C,P) on which there is a Lévy process x· deﬁned. Then Pxt , the distribution of the random variable xt under P , which is a probability measure on Rd,canbeshowntogiverisetoaspace- translation invariant Markov kernel by setting pt(z,B)=Pxt [xt ∈ B − z].

Moreover, one can prove (pt)t∈R+ to be a translation-invariant Markovian semigroup which is continuous as a map t → pt (with respect to the vague, or weak∗ topology). The converse also holds true: For every translation-invariant continuous Markovian – i.e. translation-invariant Feller – semigroup p· there is a Lévy process x· on a probability space (Ω,C,P) such that pt = Pxt for all t ≥ 0. For this, see e.g. Bauer [77]. 202 5 Hyperﬁnite Lévy Processes and Applications

As hinted at previously, one can now rephrase the Lévy-Khintchine formula: Theorem 5.1.2 (Lévy-Khintchine formula for inﬁnitesimal genera- d tors). Consider a Markovian semigroup (pt)t∈R+ on R , d ∈ N.Suppose ∗ t → pt is continuous (in the vague or weak topology). Then the semigroup p· d is space-translation invariant (in the sense that ptf(·+z)=ptf for all z ∈ R and any nonnegative measurable f : Rd → R) if and only if the inﬁnitesimal

generator of (pt)t∈R+ can be written as 1 d d : f → Ci,j ∂i∂jf + γi∂if + (f(· + y) − f) ν(dy) (5.1.2) 2 d i,j=1 i=1 R \{0}

d×d where C ∈ R+ is a symmetric d × d-matrix with nonnegative entries, γ ∈ Rd,andν is a Radon measure on Rd \{0} satisfying 1. |y|2 ν(dy) < +∞, B1(0) 2. ν[B1(0)] < +∞.

Via the Ionescu-Tulcea-Kolmogorov construction, p· is a translation-invariant semigroup if and only if there is a Lévy process x =(xt)t≥0 on a probability

space (Ω,C,P) such that Pxt = pt for all t ≥ 0.

(For this result, see e.g. [307].)

The latter formulation of the Lévy-Khintchine formula can also be seen as the link between the theory of Lévy processes and certain Dirichlet forms:

Remark 5.1.1. To all inﬁnitesimal generators of Rd-valued Lévy processes and measures m on the Borel σ-algebra of Rd, one can immediately associate a bilinear form by setting E(f,g)= −f(x) · g(x) m(dx) Rd

for all f ∈ L2(Rd,m) and g ∈ D(),whereinD() denotes the domain of in L2(Rd,m), i.e. the preimage of L2(Rd,m) under .ThisformE can be completed to become a closed coercive form under certain conditions, viz. that (1) m is p·-supermedian (in the sense that ptfdm≤ fdmfor all t>0 and nonnegative Borel-measurable f), and (2) p· is m-symmetric (which means g · ptfdm= f · ptgdmfor all m-square integrable f,g and t>0) – conversely, one can uniquely recover the corresponding infinitesimal generator from the Dirichlet form. In the framework of symmetric forms, this can be found in Fukushima, Ôshima and Takeda [183, Theorem 1.3.1]. In the setting of nonsymmetric forms, this is explained, e.g., in Ma and Röckner [270, Proposition II.4.3, final remarks in Sect. II.4.a]. 5.2 Characterizing Hyperfinite Lévy Processes 203

Analogous representation formulae can be proven for Dirichlet forms, too (cf. [271]). For closability questions of minimally defined symmetric Dirichlet forms with jump processes, we refer to Albeverio and Song [56]. See also Albeverio and Rüdiger [51] for subordination methods, in order to get symmetric Dirichlet forms that are not necessarily of diffusion type. For other work on non-diffusion forms, see, e.g., Jacob [218–220] or Jacob and Schilling [222].

5.2 Hyperﬁnite Lévy Processes: Deﬁnitions and Characterizations

Keeping the above results about Lévy processes in a standard setting in mind, we shall now introduce the key notions of the nonstandard theory of Lévy processes.

As before, the set T = {nΔt | n ∈ ∗N}, for some positive Δt ≈ 0, will serve as our time-line. Again we are working on an internal probability space (Ω,2Ω,μ), 2Ω denoting the internal power set of Ω,andμ an internal ﬁnitely- additive probability measure on Ω.

The following definition was first given by Lindstrøm [263, Definitions 1.1, 1.3].

Definition 5.2.1. [263, Definitions 1.1, 1.3] Consider an internal stochastic process X : Ω × T → ∗Rd and a hyperfinite set A ⊂ ∗Rd. • X is called a hyperfinite random walk with increments from A and transition probabilities {pa}a∈A if and only if

1. X0 =0on Ω, 2. For all t ∈ T ,theincrements ΔX0 = XΔt −X0, ··· ,ΔXt = Xt+Δt −Xt form a hyperﬁnite set of ∗-independent internal random variables, and 3. For all t ∈ T and for all a ∈ A,

μ {ΔXt = a} = pa.

• X is called a hyperﬁnite Lévy process if and only if

1. X is a hyperfinite random walk and ∗ d 2. L(μ)[ t∈T ∩Fin(∗R){Xt ∈ Fin( R )}]=1. Remark 5.2.1. Note that since T is not hyperfinite, we will never have that Ω is hyperfinite unless |A| =1(in which case X is deterministic). However, when we constrain the time horizon to a finite t ∈ T ∩ Fin(∗R),thesetofall paths up to time t will be hyperfinite. 204 5 Hyperfinite Lévy Processes and Applications

Remark 5.2.2. To any hyperﬁnite Lévy process, one can associate, in analogy to Remark 5.1.1, a bilinear, not necessarily symmetric form on ∗Rd by setting 1 ∀f,g ∈D(E) E(f,g)= f(x) · g(x) − g (x + a) pa dx ∗ d R Δt a∈A

(where D(E) depends on the expression on the right hand side via Deﬁnition 1.4.2). In order to show that this form is nonnegative, we combine the Cauchy- Bunyakovski-Schwarz inequality (transferred to the nonstandard universe) with the translation invariance of the ∗-Lebesgue measure and the fact that {a} → pa deﬁnes a probability measure on A: 1 f(x) · f (x + a) pa dx ∗Rd Δt a∈A 1 = f(x) · f (x + a) pa dx ∗ d a∈A R Δt 1/2 2 1/2 2 1 ≤ f(x) dx f (x + a) pa dx ∗ d ∗ d a∈A R R Δt ⎛ ⎞1/2 1/2 ⎜ ⎟ 2 ⎜ 2 ⎟ 1 ≤ ( ) ⎜ ( ) ⎟ a f x dx ∗ d f y dy p ∗ d ⎝ + R ⎠ Δt a∈A R a =∗Rd 1 = f(x)2 dx. Δt ∗Rd

This implies E(f,f) ≥ 0 for all f in the domain D(E) of E. So the bilinear form E is indeed nonnegative.

Conversely, if we are given a form E which is derived from such a set of increments A and transition probabilities (pa)a∈A, we can reconstruct A and ∗ d (pa)a∈A by means of elementary linear algebra in R (the corresponding operator is given explicitly in (1.1.2) of Sect. 1.1).

Thus, by studying hyperfinite Lévy processes, one also obtains results on certain nonnegative quadratic forms. We shall see in this chapter that for hyperfinite Lévy processes, the construction of standard parts and the classification of infinitesimal generators are technically less demanding than the proofs of the analogous results for general hyperfinite Markov chains. Furthermore, hyperfinite Lévy processes and their generators admit more explicit representation results: the hyperfinite Lévy-Khintchine formula (Theorem 5.4.1) and the representation of hyperfinite Lévy processes as superpositions of hyperfinite Poisson processes (Theorem 5.6.1). 5.2 Characterizing Hyperfinite Lévy Processes 205

Our goal for this section is to prove a necessary and sufficient criterion for a hyperfinite random walk to be a hyperfinite Lévy process. For this sake, we shall first of all approximate hyperfinite Lévy processes by hyperfinite Lévy processes with finite increments (in the sense that A ⊂ Fin(∗Rd)), and then prove integrability results for hyperfinite Lévy processes with finite increments. Notation 5.2.1. Unless indicated otherwise, X will always be a hyperfinite random walk with increments from A and transition probabilities {pa}a∈A. We shall also use the following abbreviation: ∗ 1 ∀k ∈ R qk = pa, Δt a∈A |a|>k whence qkΔt = μ{ΔXt ∈ Bk(0)} for any t ∈ T . In preceding chapters of this monograph, A denoted an internal operator. Since we will not be concerned with operators in this chapter, we are free to use A to denote the hyperfinite increment set.

Notation 5.2.2. By Ft,fort ∈ T , we shall denote the internal algebra generated by the internal random variables X0,...,Xt.

Notation 5.2.3. As another notational convention, we let E[Z]=Eμ[Z],for internal random variables Z, denote the expectation with respect to the m internal probability measure μ. Similarly, E[z]=EL(μ)[z],forR -valued Loeb-measurable random variables z, will denote the expectation with respect to L(μ). Analogously, E[Z|C], for an internal algebra C of a hyperﬁnite probability space Ω, denotes the conditional expectation of Z given C,and E[z|L(C)] will denote the (standard) conditional expectation of z given the Loeb extension of C.

We may adopt this notation, since we will not consider any standard Dirichlet forms (also usually denoted by E) for the rest of this chapter. ∗ Remark 5.2.3. For all u ≥ s ∈ T , ΔXu is-independent of all ΔXv with ∗ = v

Following common practice, we will often simply write “independent” when ∗-independent is really meant.

Remark 5.2.4. If u, t ∈ T with u>t, then the internal law of Xu − Xt = v

◦ Remarks 5.2.3 and 5.2.4 have prepared us for the proof of ( qk)k∈N ∈ N (R+ ∪{+∞}) being decreasing to zero whenever X is a hyperfinite Lévy process. Lemma 5.2.1. [263, Lemma 3.1] If X is a hyperfinite Lévy process, then ◦ qk ↓ 0 as k →∞. Proof. Assume the negation of the Lemma’s assertion were true, then there is an ε>0 such that qk >εfor all k ∈ N. By overflow, there must be a ∗ K ∈ N \N such that qK >ε. Using the definition of qk and recalling the fact that the increments of X are ∗-independent, we can compute the probability that X will make a change that is greater than k in norm before time t+Δt as ⎡ ⎤

t ⎣ ⎦ Δt μ {|ΔXs| >k} =(qkΔt) . s≤t

But on the other hand, one has for noninﬁnitesimal t ∈ T ,

◦ t ◦ t −εt (qkΔt) Δt ≤ (1 − εΔt) Δt = e < 1, since (1 − x/N)N −→ e−x as N →∞holds locally uniformly for all x ∈ R, which implies (1 − y/N)N ≈ e−y for all finite y ∈ ∗R and infinite N ∈ ∗R. Therefore the probability of X making a jump of infinite size before t + Δt is noninfinitesimal, contradicting the almost sure finiteness condition imposed on all hyperfinite Lévy processes X.

∗ ≤k Deﬁnition 5.2.2. Let k ∈ R>0. We deﬁne the truncated random walks X and X>k by ≤k ∗ d X : Ω × T → R , (ω,t) → ΔXs (ω) , sk ∗ d X : Ω × T → R , (ω,t) → ΔXs (ω) . sk

Remark 5.2.5. Obviously, X = X≤k + X>k on all of Ω × T ,andbothX≤k >k and X are hyperfinite random walks (with increments from {0}∪A∩Bk(0) and {0}∪A \ Bk(0), respectively). Lemma 5.2.2. [263, Lemma 3.2] If X is a hyperfinite Lévy process, then so ≤k >k ∗ are X and X for all sufficiently large finite k ∈ R+. Proof. We observe that the difference of two hyperfinite Lévy processes is again a Lévy process. Hence the first part of Remark 5.2.5 implies that we only have to verify that X>k is a hyperfinite Lévy process for sufficiently large finite k,andX − X>k = X≤k will be one as well. By the second part of 5.2 Characterizing Hyperfinite Lévy Processes 207

Remark 5.2.5, all we have to check is whether X>k will be ﬁnite for all ﬁnite times on a set of probability 1 if we choose k ∈ Fin(∗R) large enough.

By Lemma 5.2.1,wemaychoosek ∈ N such that qk is ﬁnite. Now consider, for arbitrary m>k, the internal process (k,m] ∗ d X : Ω × T → R , (ω,t) → ΔXs (ω) . s

The set of increments of X(k,m] is obviously contained in Fin(∗Rd). Hence we have (k,m] ∗ d (k,m] ∗ Xs ∈ Fin R ⊂ card sk}∈ N \ N} (5.2.1) for all t ∈ T . But on the other hand, μ{|ΔXs| >k} = qkΔt for all s ∈ T and all increments are ∗-independent, therefore we may perform the following combinatorial calculation:

L(μ)[{card {sk}∈N}] ∞ = L(μ)[{card {sk} = n}] n=0 ∞ ◦ t/Δt n t −n = (qkΔt) (1 − qkΔt) Δt . (5.2.2) n=0 n

t If Δt is ﬁnite, then we immediately get

L(μ)[{card {sk}∈N}]=1,

t and if Δt is inﬁnite, then (5.2.2) yields

L(μ)[{card {sk}∈N}] ∞ ◦ n t 1 n = −qkt ( ) n !e qkΔt n=0 Δt n ∞ ◦ ∞ 1 ◦ 1 = ◦ −qkt n n = − qkt n ◦ n =1 e t !qk e t ! qk n=0 n n=0 n

h−n −q t since (1−qkt/h) ≈ e k for all infinite h – in particular for h = t/Δt –and finite n ∈ N (this, in turn, follows from the fact that (1 − x/N)N−n −→ e−x as N →∞locally uniformly for all x ∈ R and for any n ∈ N). Thus, in 208 5 Hyperfinite Lévy Processes and Applications ! (k,m] ∗ d view of relation (5.2.1), we have proven that s≤t{Xs ∈ Fin( R )} has L(μ)-probability zero.

Summarizing the ﬁrst part of this proof, we state that on a set of (k,m] probability 1, the internal random variable Xs is ﬁnite for all s ≤ t.

Now we turn to the hyperﬁnite random walk X>k again. Note that, again by a simple combinatorial argument based on the independence and station- (k,m] >k arity of the increments of X and X , as well as on the deﬁnition of qm, one has ⎡ ⎤ " # ⎣ (k,m] >k ⎦ ◦ L(μ) Xs = Xs = μ {|ΔXs|≤m} s≤t s

◦ ◦ t/Δt ◦ −qmt − qmt = (1 − qmΔt) = e = e

(where once again we have used the locally uniform convergence of (1 − )N −→ −x →∞ ◦ −→ 0 →∞ x/N e as N for all x). But qm as m , hence (k,m] >k L(μ)[ s≤t{Xs = Xs }] −→ 1 as m →∞. Therefore the probability for (k,m] >k the assertion that for some m ∈ N, Xs = Xs for all s ∈ T ,equalsone. (k,m] Since with probability 1, Xs is finite for all s ≤ t (as was shown in the first >k part of this proof), we obtain that with probability 1, Xs is also finite for all s ≤ t.ThusX>k is not merely a hyperfinite random walk, but a hyperfinite Lévy process. This suffices to complete the proof, as was explained at the outset.

The preceding argument also yields the following

Corollary 5.2.1. [263, Corollary 3.3] Suppose X is a hyperfinite random walk such that X≤m is a hyperfinite Lévy process for all sufficiently large ◦ finite m and assume that qm −→ 0 as m →∞.ThenX is a hyperfinite Lévy process.

Proof. As was shown in the proof of Lemma 5.2.2, the convergence assertion ◦ >k qm −→ 0 as m →∞suffices to prove that X will be a hyperfinite Lévy process. But by assumption, so is X≤k.SincewehaveX = X≤k + X>k and because the sum of two hyperfinite Lévy processes also is a hyperfinite Lévy process, we conclude that X must be one as well.

We have now almost reached the end of our study of approximations of hyperfinite Lévy processes by hyperfinite Lévy processes with finite increments.

Lemma 5.2.3. [263, Proposition 3.4] If X is a hyperfinite Lévy process, t ∈ ˆ ˆ (ε,t) T finite, and ε ∈ R>0, then there exists a hyperfinite Lévy process X = X with finite increments such that 5.2 Characterizing Hyperfinite Lévy Processes 209 ⎡ ⎤ " ⎣ ˆ ⎦ μ Xs = Xs > 1 − ε. s≤t

◦ Proof. Since qk −→ 0 as k →∞by Lemma 5.2.1,wecanchoosek ∈ ◦ −q t ≤k R>0 such that e k > 1 − ε. By Lemma 5.2.2, X is a hyperfinite Lévy process and obviously has finite increments. Moreover, similarly to the proof of Lemma 5.2.2, we can show that ⎡ ⎤ " $ % # ⎣ ≤k ⎦ ◦ ◦ t/Δt L(μ) Xs = Xs = μ {|ΔXs|≤k} = (1 − qkΔt) s≤t s 1 − ε by our initial choice of k. Thus we only have to put Xˆ = X≤k to bring the proof to a close. As some of the readers might already have suspected, hyperfinite Lévy processes with finite increments get their importance from their S-integrability properties. Theorem 5.2.1. [263, Theorem 2.3] If X is a hyperfinite Lévy process with p finite increments, then the random variable |Xt| is S-integrable for all finite ∗ p ∈ R≥0 and finite t ∈ T . The proof will employ the following Lemma: Lemma 5.2.4. [263, Lemma 2.1] Consider a finite Loeb measure space ∗ d (Ω,L(A),L(μ)),andletF : Ω → R be A-measurable and internal. If p ∗ q Ω |F | dμ is finite for some finite p ∈ R>0,then|F | is S-integrable for all ∗ q ∈ R>0 with q1} ≤ |F |p dμ {|F |>1}

Now consider an A ∈Asuch that μ(A) ≈ 0. It remains to prove that | |q ≈ 0 A F dμ . Hölder’s inequality, when transferred to the nonstandard universe, yields

q/p q p 1− q |F | · 1 dμ ≤ |F | dμ · μ(A) p A A p 1− q ≤ |F | dμ ∨ 1 · μ(A) p . Ω 210 5 Hyperﬁnite Lévy Processes and Applications

1 − q 0 1 − q ≈ 0 ≈ Hence, noting that p > , but p (otherwise q p,since p is | |p | |q finite), and recalling that Ω F dμ is finite, we conclude that A F dμ must be infinitesimal.

Proof. Ushering in the proof of Theorem 5.2.1,weﬁrstremarkthatwemay assume that Xt is not almost surely zero for all t ∈ T (otherwise there is nothing to prove). Hence (adopting the convention min ∅ = ∗∞) the stopping time τk =min{t ∈ T ||Xt|≥k } , is not equal to ∗∞ everywhere for any k ∈ ∗R.

For all infinite k, τk will be almost surely infinite, since X is a hyperfinite 1 Lévy process. Therefore, by underspill, we must have μ{τk > 1} > 2 for all sufficiently large finite k. In particular, there will be a finite noninfinitesimal k which is strictly greater than all the increments of X and satisfies μ{τk > 1 −τk 1} > 2 . Fixing such a k,wesetα = E[e ]. The choice of k ensures that 1 ≈ α<1.

Inductively, we shall deﬁne a sequence of new stopping times via σ0 =0, σ1 = τk and $ %

∀n ∈ N σn =min t ∈ T t>σn−1, Xt − Xσn−1 ≥ k .

The next step is to see that the family of random variables $ %

σn − σn−1 =min s ∈ T \{0} Xs+σn−1 − Xσn−1 ≥ k ,

∗ ∗ indexed by n ∈ N,is -independent, and that each random variable σn−σn−1 will be distributed exactly like τk. Whilst this can be proven by applying the transfer principle to the corresponding standard result (which, in turn, usually is proven by conditioning on σn−1), we prefer to give a more self- ∗ contained argument here. Note that for all n ∈ N and s, s0,...,sn−1 ∈ T , " {σn − σn−1 = s}∩ {σi = si} (5.2.3) i

= Xs+sn−1 − Xsn−1 ≥ k ∩ Xu+sn−1 − Xsn−1

Ftn−1 ⊂Fv. Therefore As must be independent of Ftn−1 as well, which via (5.2.3) implies & ' & ' " " μ {σn − σn−1 = s}∩ {σi = si} = μ [As] μ {σi = si} . (5.2.4) i

Moreover, by conditioning on the random variable Xsn−1 (which can only attain a hyperﬁnite number of values) and taking advantage of Xu+sn−1 −

Xsn−1 being independent from Fsn−1 for any u ∈ T ,weseethatμ[As]= μ{τk = s}. Exploiting (5.2.4), we thereby ﬁnally arrive at & ' & ' " " μ {σn − σn−1 = s}∩ {σi = si} = μ {τk = s} μ {σi = si} i

A consequence of this result is that μ {σn − σn−1 = s} = μ [{σn − σn−1 = s}∩{σn−1 = t}] t∈T ∪{∗∞} = μ {τk = s} μ {σn−1 = t} t∈T ∪{∗∞}

= μ {τk = s}

(note that Ω is hyperﬁnite, therefore only hyperﬁnitely many of the values ∗ μ{σn−1 = t}, t ∈ T ∪{ ∞}, will be non-zero), hence σn − σn−1 has the same distribution as τk.

Reinserting this into (5.2.5) yields & ' " μ {σn − σn−1 = s}∩ {σi = si} i

Having proven that the σn − σn−1 are independent and distributed according to the law of τk,wemaynextwrite 212 5 Hyperﬁnite Lévy Processes and Applications

n # n+1 E e−σn = E e−τk E e−σn+σn−1 = E e−τk = αn+1 (5.2.7) i=1 for every n ∈ ∗N. On the other hand, k is larger than all increments of X, | − | and by deﬁnition of σn, one will always have Xσ−Δt Xσ−1

− ≤| − | + − 2 Xσ Xσ−1 Xσ Xσ−Δt Xσ−Δt Xσ−1 < k

∗ for all ∈ N. Using this estimate as well as the deﬁnition of σn,wesee that if |X(ω,t)|≥2nk for some t ∈ T and ω ∈ Ω, then we must have that t>σn(ω), for otherwise

|Xt|≤|Xσ |≤ |Xσ − X0| ≤ Xσ − Xσ

Thus, in light of (5.2.7), we gain

−σn E [e ] t n+1 μ {|Xt|≥2nk}≤μ {σn 0 and t ∈ T , ( ) E eε|Xt| = eε|Xt| dμ n∈∗N {2(n−1)k≤|Xt|<2nk} ε·2nk 2εnk t n ≤ μ {2(n − 1)k ≤|Xt|} e ≤ e e α . n∈∗N n∈∗N (We point out that, thanks to the hyperﬁniteness of Ω, the above sum n∈∗N ε|Xt| {2(n−1)k≤|Xt |<2nk} e dμ will only feature hyperﬁnitely many non-zero addends.)

◦ 2εk In particular, if we choose ε ∈ R>0 so small that e α<1 (which is always possible since ◦α<1 and k is ﬁnite), we get ( ) 1 E eε|Xt| ≤ et · e2εkα · ∈ Fin(∗R) 1 − e2εkα for all ﬁnite t ∈ T .

Finally, if we remark that for all finite p there is always a finite N0 ∈ N ε|X | p p such that e t > |Xt| on {|Xt| >N0}, we conclude that E[|Xt| ] is finite. Applying Lemma (5.2.4) completes the proof of the theorem. 5.2 Characterizing Hyperfinite Lévy Processes 213

We introduce the following quantities, which may be called normalized increment mean and normalized increment variance, respectively. Deﬁnition 5.2.3. ( ) 1 1 2 1 2 1 2 mX = E [ΔX0]= apa,σX = E |ΔX0| = |a| pa. Δt Δt a∈A Δt Δt a∈A

Lemma 5.2.5. [263, Lemma 1.2, Corollary 2.4] • For all t ∈ T and arbitrary X, ( ) 2 2 2 E |Xt| = σX · t + |mX | · t (t − Δt) .

• Moreover, if X is a hyperfinite random walk with finite increments, then 2 X will be a hyperfinite Lévy process if and only if both mX and σX are finite.

Proof. For the ﬁrst part of the Lemma, observe that whenever s = u, ΔXs and ΔXu are independent and therefore

2 2 2 E [ΔXs · ΔXu]=E [ΔXs] · E [ΔXu]=|E [ΔX0]| = |mX | Δt , whence this quantity is independent of u, s. This yields & ' ( ) 2 E |Xt| = E ΔXs · ΔXu s

We shall now prove the criterion of the second part of the Lemma.

2 2 Assume first that mX and σX are finite. Then, so is E[|Xt| ] for all finite t ∈ T bythefirstpartoftheLemma.

Next we point out that the process (Xt − mX t)t∈T is not only a hyperfinite random walk, but also a martingale, since due to the independence and stationarity of the increments, E[ΔXt − mX Δt|Ft]=E[ΔXt] − 2 mX Δt = E[ΔX0] − mX Δt =0for all t ∈ T .Furthermore,E[|Xt − mX t| ] 2 E[|Xt − mX t| ] is finite for all finite t ∈ T , since on the one hand 214 5 Hyperfinite Lévy Processes and Applications

2 E[|Xt| ] is finite for all finite t ∈ T , and on the other hand (due to the Cauchy-Bunyakovski-Schwarz inequality), one has ( ) ( ) 2 2 2 2 E |Xt − mX t| ≤ E |Xt| + |mX | t +2|mX · E [Xt]| t ( ) ≤ | |2 + | |2 2 +2| |· | [ ]| · E Xt mX t mX E Xt t 2 ≤E[|Xt|]≤1+E[|Xt| ] ( ) 2 2 2 ≤ E |Xt| (1 + 2 |mX | t)+|mX | t +2|mX | t, whilst mX is finite by assumption.

Thus, we have shown that (Xt − mX t)t∈T is an internal martingale such 2 that E[|Xt − mX t| ] is finite for any finite t ∈ T . Hence we may employ the subsequent Lemma 5.2.6 and conclude that with probability 1, Xt − mX t is finite for all finite t ∈ T . In light of the finiteness of mX , this means that with probability 1, Xt is finite for all finite t ∈ T .

In order to prove that the criterion also works in the converse direction, 2 we recall Theorem 5.2.1 to find that E[|Xt − mX t| ] is finite for all finite t ∈ T . But, being aware of the first part of the present Lemma, this can only 2 be true if both mX and σX are finite.

Lemma 5.2.6. [25, p. 119] If M is an internal martingale on an arbitrary 2 hyperfinite adapted Loeb probability space (Ω,(Ft)t∈T ,L(μ)),then(|Mt| )t∈T 2 is a submartingale, and if E[|Mt| ] is finite for all finite t ∈ T (i.e. M is a 2 λ -martingale), then there is a subset Ω0 ⊂ Ω of L(μ)-measure 1 such that Mt(ω) is finite for all ω ∈ Ω0 and finite t ∈ T .

Proof. Consider a ﬁnite t ∈ T , and assume for a contradiction that there was asetB˜ of Loeb measure p>0 such that for all ω ∈ B˜,thereisans ≤ t such that Ms(ω) is inﬁnite, i.e. we assume ⎡ ⎤ ⎣ ∗ d ⎦ p = L(μ) Ms ∈ Fin R > 0. s≤t ! ∗ Then, by ℵ1-saturation, there will be an N ∈ N\N such that μ[ s≤t{|Ms|≥ 1 N}] ≥ p − N . This can be written, when we introduce the stopping time

τ =min{s ∈ T ||Ms|≥N } ,

1 as μ{τ ≤ t}≥p − N , i.e. L(μ){τ ≤ t} > 0. Hence the process 2 (|Mt −mX t| )t∈T is the square of a martingale and therefore a submartingale. The submartingale property of squares of internal martingales follows e.g. from applying the Transfer Principle to Jensen’s inequality for conditional 5.2 Characterizing Hyperﬁnite Lévy Processes 215 expectations [77, p. 121, Satz 15.3] or from simply observing that for all martingales M, one has ( ) ( ) 2 2 ∀t ∈ TEΔ |Mt| Ft − E |ΔMt| Ft =2E [MtΔMt|Ft] =2 [ |F ] =0 Mt E ΔMt t =0

2 2 and therefore E[Δ(|Mt| )|Ft]=E[|ΔMt| |Ft] ≥ 0 for any t ∈ T .

If we apply the transferred Optional Stopping Theorem to this submartingale and the stopping times τ ∧ t ≤ t,weget ( ) ( ) 2 2 E |Mt| ≥ E |Mτ∧t| . (5.2.8)

On the other hand, ( ) 2 2 E |Mτ∧t| ≥ μ {τ ≤ t} · min |Mτ∧t| . {τ≤t} ≈0 ∗ ∈ R>0\R

2 (The minimum min{τ≤t} |Mτ∧t| will be attained as the infimum of an inter- 2 2 nal function on a hyperfinite set, and it will be infinite since |Mτ∧t| ≥ N ∈ ∗ 2 2 N \ N on {τ ≤ t}.) So, E[|Mτ∧t| ] and therefore E[|Mt| ] will be infinite, a contradiction to what was supposed in the Lemma’s statement. Thus we have brought our initial assumption to absurdity. Hence for all finite t ∈ T , ∗ d we do have the identity L(μ)[ s≤t{Ms ∈ Fin( R )}]=1.Afterchoosing ε ∈ T \ st−1{0}, we now conclude the proof by remarking that ⎡ ⎤ " ⎣ ∗ d ⎦ L(μ) Ms ∈ Fin R s∈T ∩Fin(∗R) ⎡ ⎤ " " ⎣ ∗ d ⎦ = L(μ) Ms ∈ Fin R , (5.2.9) n∈N s≤nε where the right hand side is the probability of a countable intersection of events of measure 1.

So far, we have only truncated hyperﬁnite random walks X to processes such as X≤k and X>k. We would like to consider more general truncation schemes. This also prompts us to generalize the notion of qk. 216 5 Hyperﬁnite Lévy Processes and Applications

Deﬁnition 5.2.4. For all internal B ⊂ ∗Rd, 1 νˆ [B]= pa Δt a∈A∩B

Note that qk =ˆν[Bk(0)].

Lemma 5.2.7. [263, Proposition 4.1] If X is a hyperﬁnite Lévy process and B ⊂ ∗Rd is internal such that B ∩ st−1{0} = ∅,thenνˆ[B] is ﬁnite.

Proof. Since νˆ[B ∩ Bk(0)] ≤ qk ↓ 0 as k →∞by virtue of Lemma 5.2.1, it suffices to prove that νˆ[B ∩ Bk(0)] is finite for some k ∈ N.Thuswemay assume without loss of generality that B ⊂ Bk(0) for some finite k.Ifthis k is large enough, then Lemma 5.2.2 tells us that X≤k is a hyperfinite Lévy process. On the other hand, X≤k has increments that are finite (bounded by 2 k), thus by the previous Lemma 5.2.5, σX≤k is finite. Recalling that B is internal and does not contains any infinitesimal elements by assumption, we may apply underspill to find an ε ∈ R>0 such that B ∩ Bε(0) = ∅.Then 2 1 2 1 2 1 2 2 ε νˆ[B]= ε pa ≤ |a| pa ≤ |a| pa = σX≤k Δt Δt Δt a∈A∩B a∈A∩B a∈A∩Bk(0) wherewehaveexploitedthatB ⊂ Bk(0).Sinceε is noninfinitesimal and 2 σX≤k is finite as remarked earlier in this proof, we have found a finite bound 2 2 σX≤k /ε on νˆ[B].

As previously announced, we shall study generalizations of X≤k:

Deﬁnition 5.2.5. Let Λ ⊂ ∗Rd be internal. We deﬁne the Λ-truncated random walk XΛ by Λ ∗ d X : Ω × T → R , (ω,t) → ΔXs (ω) s

We will now generalize Lemma 5.2.2:

Lemma 5.2.8. [263, Corollary 4.2] Let X be a hyperﬁnite Lévy process and assume Λ ⊂ ∗Rd to be internal. If either Λ∩st−1{0} = ∅ or Λ∩st−1{0} = ∅, then both XΛ and XΛ are hyperﬁnite Lévy processes.

Corollary 5.2.2. If X is a hyperfinite Lévy process, then X≤k and X>k are hyperfinite Lévy processes for all noninfinitesimal k.

Proof (Proof of Lemma 5.2.8). The proof strongly resembles the one of Lemma 5.2.2. It suffices to treat the case where Λ ∩ st−1{0} = ∅ and prove that XΛ is a hyperfinite Lévy process. For, X −XΛ = XΛ, and the difference 5.3 Hyperfinite Lévy Processes: Standard Parts 217

of two hyperfinite Lévy processes is again a hyperfinite Lévy process. But if Λ ∩ st−1{0} = ∅, then Lemma 5.2.7 yields that νˆ[Λ] is finite. Hence the value νˆ[Λ] may play exactly the rôle which was filled by qk in the proof of Lemma Λ 5.2.2. Next define, for finite m, Λm = Λ ∩ Bm(0).ThenX m can play the rôle that was filled by X(k,m] in the proof of Lemma 5.2.2. With these mod- Λ Λ Λ >k ifications (using νˆ[Λ] instead of qk, and replacing X , X , X m by X , X≤k, X(k,m], respectively), we can now copy the proof of Lemma 5.2.2 to see Λm that there is a set of probability 1 on which Xt is finite for all finite t ∈ T . Thus, XΛm already is a hyperfinite Lévy process. But XΛm =(XΛ)≤m.By means of Corollary 5.2.1, we deduce that XΛ is a hyperfinite Lévy process as well.

Now we are ready to state and prove the central result of this section.

Theorem 5.2.2. [263, Theorem 4.3] The hyperfinite random walk X is a hyperfinite Lévy process if and only if all of the following conditions are satisfied: ∗ −1 1 ∗ d 1. For all k ∈ Fin( R) \ st {0}, apa ∈ Fin( R ). Δt |a|≤k ∈ Fin(∗R) 1 | |2 ∈ Fin(∗Rd) 2. For all k , Δt |a|≤k a pa . ◦ 1 3. limk→∞ ( Δt |a|>k pa)=0. Proof. In case X is a hyperfinite Lévy process, we may apply Lemma 5.2.8 and see that X≤k is a hyperfinite Lévy process – and even one with increments that are finite (bounded by k) – for all finite noninfinitesimal k. Now Lemma 1 5.2.5 already yields the finiteness of both Δt |a|≤k apa – thereby proving 1 | |2 the first condition in the statement of the theorem – and Δt |a|≤k a pa for 1 all finite noninfinitesimal k.Sincek → Δt |a|≤k apa is increasing in k,this suffices to prove the second condition as well. The third condition, finally, is simply the conclusion of Lemma 5.2.1.

To prove that the converse implication also holds, we note that by Lemma 5.2.5, the first two conditions in the statement of the theorem imply that X≤k must be a hyperfinite Lévy process for all finite noninfinitesimal k.Nowthe third condition entitles us to apply Corollary 5.2.1 – which readily assures us of X indeed being a hyperfinite Lévy process.

5.3 Standard Parts of Hyperﬁnite Lévy Processes

d Deﬁnition 5.3.1. Let F : T → R be internal, and consider s ∈ R≥0 and c ∈ ∗ d R .Thenc is called the S-right limit of F at s, denoted c = S- lim◦t↓r F (t), whenever for all ε ∈ R>0 there exists a real number δ ∈ R>0 such that |F (t) − c| <εfor all t ∈ T ∩ ∗[s, s + δ). 218 5 Hyperﬁnite Lévy Processes and Applications

Analogously, c is called the S-left limit of F at s, denoted c = S- lim◦t↑r F (t), whenever for all ε ∈ R>0 there is some δ ∈ R>0 such that |F (t) − c| <εfor all t ∈ T ∩ ∗(s − δ, s].

The value c is called an S-one-sided limit of F at s if it is either the S-left limit or the S-right limit of F at s.

Remark 5.3.1. All S-one-sided limits are obviously unique – this would even hold if F was taking values in ∗Y , Y being an arbitrary Hausdorﬀ space, rather than ∗Rd.

In order to establish the existence of one-sided limits for almost all paths of a hyperﬁnite Lévy process, we shall use the following well-known result from the theory of λ2-martingales:

Theorem 5.3.1. [25, Proposition 4.2.10] If M is an internal ∗R-valued 2 (Ft)t∈T -martingale such that E[Mt ] is ﬁnite for all ﬁnite t ∈ T , then almost all of its paths M·(ω) have one-sided limits for all s ∈ R≥0.

Proof. Since countable intersections of events of probability 1 have again probability 1, it suﬃces to show that for all ﬁnite R ∈ T , almost all of the paths M·(ω), ω ∈ Ω,haveS-one-sided limits for all s ∈ R ∩ [0,R].

The next part of the proof resembles, to some extent, the proof of Doob’s martingale convergence theorems via the “upcrossing inequalities”. For, by Lemma 5.2.6 we know that almost all paths of M remain finite in finite time, hence the conclusion of the Lemma can only fail if with positive! probability, the paths of M are “infinitely oscillating”, i.e. if the event E = a,b∈Q Ea,b, wherein $ * +% N Ea,b = ω ∈ Ω ∃(tn)n∈N ∈ T ∀n ∈ N Mt2n−1 (ω) ≤ a

∗ Deﬁne inductively the following N0-sequence of stopping times: τ0 =0 and

∗ ∀n ∈ N τ2n−1 =min{t ∈ T | t>τ2n−2,Mt ≤ a}∧R, τ2n =min{t ∈ T | t>τ2n−1,Mt ≥ b}∧R.

Obviously, (τn(ω))n∈∗N is strictly increasing and takes values in T ∩[0,R), for all ω ∈ Ω, until it hits R (where it will remain henceforth). Therefore, if we let γ = R/Δt, then we must have τγ (ω)=R for all ω ∈ Ω. On the other ∗ hand, the transferred Optional Stopping Theorem yields that (Mτn )n∈ N is a martingale and hence the identity 5.3 Hyperﬁnite Lévy Processes: Standard Parts 219 * + * +

E Mτ Mτ − Mτ Fτ = Mτ E Mτ − Mτ Fτ =0 n n+1 n n n n+1 n n =0

∗ 2 2 for all n ∈ N follows. This in turn implies, via (Mτn+1 − Mτn ) = Mτn+1 − 2 Mτn − 2Mτn (Mτn+1 − Mτn ),therelation (* + ) 2 2 2 E Mτn+1 − Mτn Fτn = E Mτn+1 − Mτn Fτn for arbitrary n ∈ ∗N,fromwhichweobtain & ' * + 2 2 2 2 2 E MR = E Mτγ = E M0 + Mτn+1 − Mτn n<γ & ' * + 2 2 = E M0 + Mτn+1 − Mτn . (5.3.1) n<γ

On the other hand, by our deﬁnition of Ea,b and (τn)n∈∗N,wehave

* +2 2 Mτn+1 − Mτn ≥|b − a| on Ea,b for infinitely many n<γ(i.e. for all n ∈ N0). Equation (5.3.1) hence shows us 2 2 2 that E[MR ] ≥|b−a| L(μ)[Ea,b]·n for all n ∈ N.ThusE[MR ] becomes infinite by our previous assumption of Ea,b having positive probability. On the 2 other hand, E[Mt ] was assumed to be finite for all finite t ∈ T in the statement of the Lemma! This contradiction completes the proof of the theorem.

Lemma 5.3.1. [263, Proposition 6.3] If X is a hyperﬁnite Lévy process, then almost all paths of X have one-sided limits for all s ∈ R≥0.

Proof. First assume X to have finite increments. In the proof of Lemma 5.2.5, we have already seen that due to the independence and stationarity of the increments, E[ΔXt − mX Δt|Ft]=E[ΔXt] − mX Δt = E[ΔX0] − mX Δt =0 for all t ∈ T , hence (Xt − mX t)t∈T is a martingale. Furthermore, Lemma 5.2.5 makes sure that mX is finite whence (Xt − mX t)t∈T is a hyperfinite Lévy process with finite increments. By means of Theorem 5.2.1 we see that 2 E[|Xt − mX t| ] is finite for all finite t ∈ T .

Thus, Theorem 5.3.1 may be applied to the components of the ∗Rd-valued martingale (Xt − mX t)t∈T in order to see that the components of (Xt − mX t)t∈T , and therefore (Xt − mX t)t∈T itself, have S-one sided limits for all paths in a set of Loeb probability 1 – which completes the proof for the case where X has ﬁnite increments. 220 5 Hyperﬁnite Lévy Processes and Applications

If X does not have finite increments, we simply refer to Lemma 5.2.3 (stochastic approximation of hyperfinite Lévy processes by hyperfinite Lévy processes with finite increments) and use the conclusion of the present Lemma for hyperfinite Lévy processes with finite increments: Let us choose a t0 ∈ −1 T \ st {0}. Then for all n ∈ N and ε ∈ R>0, Lemma 5.2.3 provides us with a hyperfinite Lévy process with finite increments Xˆ (n,ε) such that ⎡ ⎤ " ⎣ ˆ (n,ε) ⎦ μ Xt = Xt > 1 − ε.

t≤nt0 ! ˆ (n,ε) { s = s } 1 ∈ N Hence, ε∈Q>0 s≤nt0 X X has Loeb probability for all n and therefore, ⎡ ⎤ " " ⎣ ˆ (n,ε) ⎦ L(μ) Xt = Xt =1.

n∈N ε∈Q>0 t≤nt0

Hence there exists a set Ω1 ⊂ Ω of Loeb probability 1 such that for all ˆ finite t1 ∈ T there exists a hyperfinite Lévy process X with finite increments ˆ satisfying Xt(ω)=Xt(ω) for all ω ∈ Ω1 and t ∈ T ∩ [0,t1].Sincewehave already shown in the first part of this proof that Xˆ has one-sided limits almost surely, we are done.

Whenever one-sided limits exist, we can introduce a well-deﬁned standard part:

Deﬁnition 5.3.2. Assume F : T → ∗Rd to have one-sided limits for all s ∈ R≥0. Then we can deﬁne the (right) standard part of F , denoted st(F ), via ∀s ∈ R≥0 st(F )(s)=S- lim F (t). ◦t↓s

It is an easy exercise to show the following result:

∗ d Lemma 5.3.2. If F : T → R has one-sided limits for all s ∈ R≥0,then d st(F ):R≥0 → R is right-continuous with left limits.

Now we can deﬁne the standard part of a hyperﬁnite Lévy process.

Definition 5.3.3. If X is a hyperfinite Lévy process, the standard part of =( ) st( ) X, denoted by x xs s∈R≥0 or X , is defined as

xs(ω)=st(X·(ω)) (s) for all ω ∈ Ω such that X·(ω) has one-sided limits on all of R≥0. 5.3 Hyperﬁnite Lévy Processes: Standard Parts 221

Remark 5.3.2. In light of Lemma 5.3.1, st(X·(ω)) is well-deﬁned for almost ∈ =( ) all ω Ω, making the process x xs s∈R≥0 well-deﬁned on a set of probability 1. Lemma 5.3.2, in addition, implies that all the paths of x =st(X) are right-continuous with left limits.

In order to prove that st(X) is indeed a Lévy process, we will need the following result, which may be viewed as stating the S-continuity in probability of t → Xt:

Lemma 5.3.3. [263, Lemma 6.4] If X is a hyperﬁnite Lévy process, s ∈ R≥0 −1 ◦ and t ∈ T ∩ st {s},then Xt = xs holds L(μ)-almost surely.

Proof. First we prove that it is suﬃcient to show that

∗ ∗ ∀t ∈ T ∩ Fin( R) ∀ε ∈ R>0 ∃δX (ε, t) ∈ R>0 ∀u ∈ T ∩ Fin( R) (5.3.2) |t − u| <δX (ε, t) ⇒ . μ {|Xt − Xu|≥ε} <ε

In order to convince ourselves of the suﬃciency of assertion (5.3.2), note N ◦ that whenever (sn)n ∈ T such that sn ↓ s ∈ R≥0 as n →∞, then one ◦ will have Xsn −→ xs L(μ)-almost surely as n →∞by the deﬁnition of S- ◦ right limits. Hence also Xsn −→ xs in L(μ)-probability as n →∞.Onthe other hand, a consequence of formula (5.3.2) is that for all t ∈ T ∈ st−1{s}, ◦ ◦ Xsn −→ Xt in L(μ)-probability as n →∞. Since limits in probability are ◦ almost surely unique, we conclude that Xt = xs L(μ)-almost surely.

Hence what remains to show is the validity of the formula (5.3.2). Using Lemma 5.2.3, we will first show that if we know (5.3.2) for all hyperfinite Lévy processes X with finite increments, then it will also follow for arbitrary ∗ hyperfinite Lévy processes X:Givent ∈ T ∩ Fin( R) and ε ∈ R>0,letus assume 1 ∈ T without loss of generality and consider the hyperfinite Lévy process with finite increments Xˆ = Xˆ (ε/2,t+1) – which was constructed in ε Lemma 5.2.3 – and find the corresponding δ = δXˆ (ε/2,t+1) ( 2 ,t) as in formula (5.3.2). Then ⎛ ⎞ |t − u| <δ⇒ ⎜ ⎟ ∗ ⎜ ˆ − ˆ ≥ 2 ε ⎟ ∀u ∈ T ∩ Fin( R) μ Xt Xu ε/ < 2 , ⎝ (! ) ⎠ ˆ ε μ v≤t+1 Xv = Xv < 2 which implies ∗ |t − u| <δ∧ 1 ⇒ ∀u ∈ T ∩ Fin( R) ε ε . μ {|Xt − Xu|≥ε/2} < 2 + 2 = ε 222 5 Hyperfinite Lévy Processes and Applications

Now all that is left to prove is that (5.3.2)holdsforX being a hyperﬁnite Lévy process with ﬁnite increments. Consider t, u ∈ T with t

2 whenever |u−t| <δ.SincebothσX and mX are finite – due to Lemma 5.2.5 and the finiteness of the increments X –, we can choose δ ∈ R>0 so small 2 2 2 3 that σX δ + |mX | · δ <ε and we are done. In order to be able to justify the choice of the term “hyperfinite Lévy process”, we should expect the subsequent theorem to hold (which we are now ready to prove, based on Lemma 5.3.3). Theorem 5.3.2. [263, Theorem 6.6] If X is a hyperfinite Lévy process, then x =st(X) is a Lévy process. Proof. We simply verify that st(X) has all the defining features of a Lévy process, as laid down in Definition 5.1.1: • Almost all the paths of x =st(X) are right-continuous with left limits by Remark 5.3.2. • Because of the previous Lemma 5.3.3, x0 =0almost surely. • In order to show the independence of the increments of x =st(X), we will mimic the proof given by Anderson for the fact that the ∗-independence of ∗R-valued internal random variables implies the S-independence of their standard parts [61, Lemma 20]. (Essentially, the same argument can be found in the volume by Albeverio, Fenstad, Høegh-Krohn, and Lind- strøm [25].) Suppose, for this purpose, that we are given real numbers d s0 < ···

$ % n"−2 ◦ ◦ ◦ B = Xtn − Xtn−1 ∈ An−1 ∩ { Xti ∈ Ai} i=1 $ % n"−2 ◦ ◦ ∗ ◦ ∗ = Xtn − Xtn−1 ∈ st ( An−1) ∩ { Xti ∈ st ( Ai)} i=1 5.3 Hyperﬁnite Lévy Processes: Standard Parts 223 ⎛ ⎞ ⎜, - , -⎟ " ⎜ * + 1 n"−2 1 ⎟ = ⎜ ∗ − ∗ ≤ ∩ ∗ ( ∗ ) ≤ ⎟ ⎜ d Xtn Xtn−1 , An−1 d Xti , Ai ⎟ ⎝ m m ⎠ m∈N i=1 =:Cm =:Dm

(wherein d is the function such that d(x, A),forx ∈ Rd and compact A ⊂ Rd denotes the minimal distance of x from A), since st(∗A)=A for all compact A ⊂ Rd – cf. Proposition A.1.1 or Albeverio et al. [25, Proposition 2.1.6 (iv)’]. Note that due to Remark 5.2.3, we also know that

Xtn − Xtn−1 is independent of Ftn−1 , hence Cm is independent of Dm for all m ∈ N and therefore

◦ ∀m ∈ N L(μ)[Cm ∩ Dm]= μ [Cm] μ [Dm] .

Using that both (Cm)m∈N and (Dm)m∈N are increasing,

L(μ)[B]= inf L(μ)[Cm ∩ Dm]= inf L(μ)[Cm] · L(μ)[Dm] m∈N m∈N

=infL(μ)[Cm] · inf L(μ)[Dm] m∈N m∈N & ' & ' " " = L(μ) Cm · L(μ) Dm m∈N m∈N & ' $ % n"−2 ◦ ◦ ◦ = L(μ) Xtn − Xtn−1 ∈ An−1 L(μ) { Xti ∈ Ai} (5.3.3) i=1

(where again we have used that st(∗A)=A for all compact A ⊂ Rd). On the other hand, Lemma 5.3.3 yields & ' $ % n"−2

L(μ)[B]=L(μ) xsn − xsn−1 ∈ An−1 ∩ {xsi ∈ Ai} i=1

as well as $ % $ % ◦ ◦ L(μ) Xtn − Xtn−1 ∈ An−1 = L(μ) xsn − xsn−1 ∈ An−1

and & ' & ' n"−2 n"−2 ◦ L(μ) { Xti ∈ Ai} = L(μ) {xsi ∈ Ai} . i=1 i=1 Via (5.3.3), we conclude that 224 5 Hyperﬁnite Lévy Processes and Applications & ' $ % n"−2

L(μ) xsn − xsn−1 ∈ An−1 ∩ {xsi ∈ Ai} i=1 & ' $ % n"−2

= L(μ) xsn − xsn−1 ∈ An−1 L(μ) {xsi ∈ Ai} . i=1

Since this last equation holds for arbitrary choice of compact A1,...,An−1, d it must even hold for all Borel sets A1,...,An−1 ⊂ R , thereby completing

the proof for the independence of xsn − xsn−1 from Fsn . • Using the same notation as in the previous bullet point, what remains to

show is that the law of xsn −xsn−1 under L(μ) equals the law of xsn−sn−1 − x0 under L(μ).

d We have already seen that for all compact An−1 ⊂ R , $ % L(μ) xs − xs ∈ An−1 =infL(μ)[Cm] . (5.3.4) n n−1 m∈N

Since this holds for arbitrary sn >sn−1,wemayreplacesn by sn − sn−1 and sn−1 by 0,inordertoget(viatn − tn−1 ≈ sn − sn−1)theidentity $ % L(μ) xs −s − x0 ∈ An−1 =infL(μ)[Cm] (5.3.5) n n−1 m∈N

= {∗ ( − ∗ ) ≤ 1 } ∈ N wherein Cm d Xtn −tn−1 X0, An−1 m for all m . But since X is a hyperﬁnite Lévy process, we can simply recall Remark 5.2.4 and obtain μ[Cm]=μ[Cm] for all m ∈ N. This readily yields, via (5.3.4)and (5.3.5): $ % $ %

L(μ) xsn − xsn−1 ∈ An−1 = L(μ) xsn −sn−1 − x0 ∈ An−1

d for every compact An−1 ⊂ R and therefore even for all Borel sets An−1 ⊂ d R . Hence xsn − xsn−1 and xsn−sn−1 − x0 have the same law under L(μ) for all sn−1 0.

5.4 Lindstrøm’s Hyperﬁnite Lévy-Khintchine Formula

Definition 5.4.1. Let X be a hyperfinite Lévy process. A positive infinites- −1 ∗ imal η ∈ st {0}∩ R>0 is called a splitting infinitesimal if and only if 1 2 ∀ε ≈ 0 |a| pa ≈ 0 (5.4.1) Δt a∈A |a|∈[η,ε] 5.4 Hyperfinite Lévy-Khintchine Formula 225

Remark 5.4.1. Underspill and overspill show that the deﬁning formula (5.4.1) for a splitting inﬁnitesimal is equivalent to 1 2 S- lim |a| pa =0. ◦c↓0 Δt a∈A |a|∈[η,c]

Lemma 5.4.1. [263, p. 528] If X is a hyperfinite Lévy process, then there exists a positive infinitesimal η0 ≈ 0 such that all positive infinitesimals η ≥ η0 are splitting infinitesimals. 1 2 Proof. Since for all ε the function η → a∈A |a| pa is decreasing, we Δt |a|∈[η,ε] only have to find one splitting infinitesimal. To construct such an infinitesimal, we observe that by the second condition in Theorem 5.2.2,thequantity 1 2 a∈A |a| pa is not only increasing in η, but also finite for all finite η. Δt |a|∈[0,η) Now put ⎧ ⎛ ⎞ ⎫ ⎨⎪ ⎬⎪ ◦ ⎜ 1 ⎟ =inf ⎝ | |2 ⎠ | ∈ ∗R ◦ 0 β ⎪ a pa ζ >0, ζ> ⎪ ⎩ Δt a∈A ⎭ |a|∈[0,ζ] as well as ⎧ ⎫

⎨⎪ ⎬⎪ ∗ 1 2 −1 B = η ∈ R>0 |a| pa >β− η ⊇ st (R>0) . ⎪ Δt ⎪ ⎩ a∈A ⎭ |a|∈[0,η)

Note that B ∩ ∗(0, 1] is internal and contains ∗(0, 1] \ st−1{0}, therefore there must be an inﬁnitesimal element η ∈ B ∩ ∗(0, 1] ⊂ B (by underspill).

We shall now prove that this η is indeed a splitting infinitesimal. 1 2 ◦ 1 Since ε → a∈A |a| pa is increasing, we readily know ( a∈A Δt |a|∈[0,ε] Δt |a|∈[0,ε] 2 |a| pa) ≤ β for all infinitesimal ε. Hence we will have for any infinitesimal ε ≥ η an infinitely small hyperreal δ ∈ st−1{0} such that 1 2 1 2 |a| pa >β− η ≥ |a| pa − η + δ Δt a∈A Δt a∈A |a|∈[0,η) |a|∈[0,ε] and thus 1 2 1 2 1 2 0 ≈ η − δ> |a| pa − |a| pa = |a| pa ≥ 0. Δt a∈A Δt a∈A Δt a∈A |a|∈[0,ε] |a|∈[0,η) |a|∈[η,ε]

226 5 Hyperﬁnite Lévy Processes and Applications

The name of a splitting infinitesimal alludes to the rôle it plays in decom- posing hyperfinite Lévy processes into a sum of a drift term, a hyperfinite Lévy martingale with infinitesimal increments and a so-called hyperfinite jump martingale (i.e. a hyperfinite Lévy martingale where the contribution of the infinitesimal increments is negligible), cf. Lindstrøm [263, Theorem 5.3].

Therefore, we would expect that it has some importance in formulating and proving a hyperfinite analogon to the Lévy-Khintchine formula. The d previously defined νˆ induces a measure νx on R \{0} which shall play the rôle of the Lévy measure in the hyperfinite Lévy-Khintchine formula. In order to prove the said formula, we need some facts about this measure νx,and also some information about candidates for the hyperfinite analogue of the covariance matrix. Definition 5.4.2. Let X be a hyperfinite Lévy process. Then ( ) −1 ∗ d νx[C] = lim L (ˆν) st (C) ∩ x ∈ R ||x|≥ε Rε↓0

for all those C for which the right-hand side is well-defined. = d [ ] ⊂ Rd st−1( )= Remark 5.4.2. 1. For all compact cubes C i=1 ai,bi , C d ∗ 1 1 Ω Ω n∈N i=1 (ai − n ,bi + n ) ∈ L(2 ), 2 denoting the internal power set of Ω. Hence νx[C] is defined whenever C is a compact cube. 2. Thanks to the Carathéodory extension theorem, νx can be extended to a completed measure on the Borel σ-algebra of Rd. 3. By Lemma 5.2.7, L(ˆν)[st−1(C) ∩{x ∈ ∗Rd ||x|≥ε}] is finite for all d ε ∈ R>0 and all Borel sets C ⊂ R .

∗ d×d Deﬁnition 5.4.3. Deﬁne CX ∈ R via 1 ∀i, j ∈{1,...,d} (CX )i,j = aiajpa. Δt a∈A

Lemma 5.4.2. [263, Proposition 7.1, Lemma 7.4] Let X be a hyperﬁnite Lévy process. Then νx satisﬁes 2 | | x( ) +∞ 1. B1(0) y ν dy < ; 2. νx[B1(0)] < +∞; 3. νx{0} =0, and CX has the following properties: ∗ d 4. CX is symmetric and for all y ∈ R ,

1 2 2 2 y · (CX )y = |a · y| pa ≤ σX |y| . Δt

∗ d×d Thus, CX is positive semi-deﬁnite, which also implies C ∈ R≥0 . 5.4 Hyperﬁnite Lévy-Khintchine Formula 227

Proof. 1. By construction of νx (cf. Remark 5.4.2) and due to the fact that ◦ Xdν= XdL(ν) for all internal measures ν and S-integrable X (cf. [267, Theorem 3] and [61, Theorem 6]), we have for all ε ∈ R>0, 2 ◦ 2 |y| νx(dy) ≤ |z| L (ˆν)(dz) ∗ B1(0)\Bε(0) (B1+ε(0)\Bε/2(0)) ◦ = |a|2νˆ(da) ∗(B (0)\B (0)) 1+ε ε/2 ◦ 2 ≤ |a| pa < +∞, a∈A |a|≤2

where in the last step we have used Theorem 5.2.2. Combining this with 2 2 |y| νx(dy) = lim |y| νx(dy) ε↓0 B1(0) B1(0)\Bε(0)

(which follows from monotone convergence since νx{0} =0), we arrive at 2 ◦ 2 | | x( ) ≤ | | a +∞ B1(0) y ν dy |a|≤2 a p < . 2. Similarly, the deﬁnition of νx implies , - 1 (0) ≤ (ˆ) ∈ ∗Rd | |≥ = | | νx B1 L ν y y 2 a pa 1 |a|≥ 2

which is ﬁnite by Theorem 5.2.2. −1 3. Since st {0}\Bε(0) = ∅ for all ε ∈ R>0, this part of the Lemma is immediate from the deﬁnition of νx. 4. Since aiaj = ajai for all i, j, the symmetry of CX is trivial. The formula follows from rearranging terms and using the Cauchy-Bunyakovski-Schwarz inequality:

d d 1 y · CX y = yi (CX )i,j yj = yi aiajpayj i,j=1 i,j=1 Δt a∈A d 1 1 2 = pa yiaiajyj = pa |a · y| Δt i,j=1 Δt a∈A a∈A =(y·a)(a·y) 1 2 2 ≤ pa|a| |y| . Δt a∈A 2 =σX 228 5 Hyperﬁnite Lévy Processes and Applications

Lemma 5.4.3. [263, Proposition 7.2] Let X be a hyperfinite Lévy process. Consider an internal function F : ∗Rd → R, and assume F to be S-continuous in a for all a ∈ Fin(∗Rd) \ st−1{0}. Suppose, furthermore, that there exists 2 ∗ d some C ∈ R>0 such that |F (a)|≤C · (|a| ∧ 1) for all a ∈ R .Then ◦ ◦ F (y) νx(dy)= F (a) L (ˆν)(da) < +∞ d R Bη(0) for all splitting infinitesimals η. 2 1 2 Proof. Firstnotethatboth |a| νˆ(da)= |a| pa and Bη(0) Δt |a|>η νˆ[B1(0)] are finite due to Theorem 5.2.2. Then, by employing the estimates on F and the fact that ◦ Xdν= XdL(ν) for all internal measures ν and S-integrable X (cf. [267, Theorem 3] and [61, Theorem 6]), one gets

◦ ◦F (a) L (ˆν)(da)= F (a)ˆν(da) Bη (0) Bη (0) ◦ ( ) 2 ◦ ≤ C · |a| νˆ(da)+C · νˆ B1(0) Bη (0) and we have proven ◦F (a) L(ˆν)(da) < +∞. Bη (0) On the other hand, if we assume F to be nonnegative, we see that for all n ∈ N,

◦ ◦ F (a)ˆν(da) ≤ F (y) νx(dy) B (0)\B (0) B (0)\B (0) n− 1 2/n n 1/n n ◦ ≤ F (a)ˆν(da). (5.4.2) B (0)\B (0) n+ 1 1/2n n

Next, using the estimates for F on B1(0) and the assumption that η is a splitting inﬁnitesimal, we get F (a)ˆν(da) − F (a)ˆν(da) B (0)\B (0) B (0)\B (0) n+ 1 1/2n n+ 1 η n n ≤ 2C |a|2 νˆ(da) ≈ 0, B1/2n(0)\Bη (0) ( )ˆ( ) ≈ ( )ˆ( ) hence B (0)\B (0) F a ν da B (0)\B (0) F a ν da ,andsimi- n+ 1 1/2n n+ 1 η n n larly, B (0)\B (0) F (a)ˆν(da) ≈ B (0)\B (0) F (a)ˆν(da) for all n ∈ N. n− 1 2/n n− 1 η n n 5.4 Hyperﬁnite Lévy-Khintchine Formula 229

◦ If we combine these results with the fact that Xdν= XdL(ν) for all internal measures ν and S-integrable X (cf. [267, Theorem 3] and [61, Theorem 6]), the estimates (5.4.2)canbewritten ◦ ◦ F (a) L (ˆν)(da) ≤ F (y) νx(dy) B (0)\B (0) B (0)\B (0) n− 1 η n 1/n n ≤ ◦F (a) L (ˆν)(da). (5.4.3) B (0)\B (0) n+ 1 η n

Using the standard monotone convergence theorem for the Loeb measure L(ˆν) and for νx, we ﬁnd that ◦ ◦ ◦ F (a) L (ˆν)(da) ≤ F (y) νx(dy) ≤ F (a) L (ˆν)(da), d Bη (0) R Bη (0) hence

◦ ◦ ◦ F (y) νx(dy)= F (a) L (ˆν)(da)= F (a)ˆν(da). d R Bη (0) Bη (0)

Hence we have proven the assertion of the Lemma for nonnegative F . Needless to say, the Lemma’s assertion in full generality follows immediately from applying this partial result to the nonnegative functions Fχ{F>0} and (−F )χ{F<0}, and afterwards exploiting the linearity of the integral.

We conclude this section by stating and proving the hyperﬁnite Lévy- Khintchine formula:

Theorem 5.4.1. [263, Theorem 8.1] If X is a hyperfinite Lévy process, k ∈ ∗ R>0 finite and noninfinitesimal and η a splitting infinitesimal for X,then for all finite y ∈ ∗Rd and any finite t ∈ T , · − t · + ity mk 2 y Cηy E eiy·Xt ≈ exp , eiy·a − 1 − iy · aχ (a) νˆ(da) Bη(0) Bk (0) wherein mk = mX≤k and Cη = CX≤η .

Proof. First note that due to Taylor’s Theorem, we can ﬁnd for all N ∈ N a ∗ ∗ and a ∈ A ∩ Bη(0) an internal S-bounded function ξ0 : BN (0) → R such that 1 ∀ ∈ ∗ (0) iy·a − 1 − · = − ( · )2 + a( ) | |2 y BN e iy a 2 y a ξ0 y Δt . 230 5 Hyperﬁnite Lévy Processes and Applications

a (Since A ∩ Bη(0) is hyperfinite, one can choose the functions ξ0 , a ∈ A ∩ a Bη(0), simultaneously whence {ξ0 } is internal.) If we set ξ0(y)= a∈A∩Bη(0) a ∗ d |a|≤η ξ0 (y) for all y ∈ R and recall that |a|≤η pa ≤ 1,wehavethus ∗ ∗ constructed an internal S-bounded function ξ0 : BN → R such that * + 1 ∀ ∈ ∗ (0) iy·a − 1 − · = − ( · )2 + ( ) | |2 y BN e iy a pa 2 y a pa ξ0 y Δt . |a|≤η |a|≤η (5.4.4) ∗ d Let us now fix a finite y ∈ R and put ξ = ξ0(y).Usingtheformulafor y · (CX≤η )y of Lemma 5.4.2,(5.4.4) then becomes * + 1 iy·a − 1 − · = − · + | |2 e iy a pa 2y CηyΔt ξ Δt . (5.4.5) |a|≤η

On the other hand, by deﬁnition of mk = mX≤k ,wehaveiy · mkΔt = E[iy · ΔX≤k]. This allows us to perform the following calculation: E eiy·ΔX0 =1+E iy · ΔX≤k + E eiy·ΔX0 − 1 − iy · ΔX≤k iy·ΔX0 ≤k =1+iy · mkΔt + E e − 1 − iy · ΔX iy·a =1+iy · mkΔt + e − 1 − iy · aχ (a) Bk(0) a∈A iy·a =1+iy · mkΔt + Δt e − 1 − iy · aχ (a) νˆ(da) Bk (0) B (0) η + eiy·ΔX0 − 1 − iy · aχ (a) Bk(0) |a|≤η

(where we have exploited that νˆ{a} = pa/Δt). On combining this result with identity (5.4.5), we obtain

2 E eiy·ΔX0 =1+RΔt + ξ |Δt| , (5.4.6) where iy·a 1 R = iy · mk + e − 1 − iy · aχ (a) νˆ(da) − y · Cηy. Bk(0) 2 Bη(0) The ∗-integral (eiy·a − 1 − iy · aχ (a))ˆν(da) can be shown to be Bη (0) Bk(0) ﬁnite if we observe that the integrand is dominated according to 5.4 Hyperﬁnite Lévy-Khintchine Formula 231

∀a ∈ A eiy·a − 1 − iy · aχ (a) ≤ eiy·a +1 + y · aχ (a) Bk(0) Bk(0) =1

≤ 2+|y|· aχB (0)(a) k ≤k ≤ 2+k|y|∈Fin (∗R)

(where, of course, the Cauchy-Bunyakovski-Schwarz inequality was used) and

◦ iy·a i◦y·◦a ◦ ◦ ∀a ∈ A ∩ Bk∧1(0) e − 1 − iy · aχB (0)(a) = e − 1 − i y · a k ∞ n ∞ n ∞ n (i◦y · ◦a) (|◦y||◦a|) |◦y| = = ≤|◦ |2 ! ! a ! n=2 n n=2 n n=2 n 2 ◦ ≤|◦a| e| y| which can be combined to get 1 * + iy·a − 1 − · ( ) ≤ 2 |y| ∨ (2 + | |) | |2 ∧ 1 e iy aχB (0) a 2 e k y a k (k ∧ 1) ∈ ( iy·a − 1 − for all a A. Then, by Lemma 5.4.3, we conclude that Bη (0) e iy · aχ (a))ˆν(da) must be finite, too. Also, Cη is finite by Lemma 5.4.2, Bk(0) therefore we obtain that R must be finite as well.

Next, we recall that since the increments of X are all ∗-independent and identically distributed, we have

( ) # iy·X iy· ΔX iy·ΔX iy·ΔX t/Δt E e t = E e s

(in the last step we have once again used that (1 + x/N)N −→ ex locally uniformly in x as N →∞). But since ξ = ξ0(y) is finite, one also has tξΔt ≈ 0 and hence etR+tξΔt ≈ etR. Inserting this into (5.4.7)givesus◦E[eiy·Xt ]=etR, and resubstituting R completes the proof. 232 5 Hyperfinite Lévy Processes and Applications 5.5 Lindstrøm’s Hyperfinite Representation Theorem for Lévy Processes

In this section, we shall prove what may be regarded as the central result of this chapter: that for each triple consisting of a drift vector γ ∈ Rd,a d×d symmetric matrix C ∈ R≥0 and a Lévy measure ν, there is a corresponding hyperﬁnite Lévy process X such that its standard part x =st(X) has covariance matrix C,driftγ and Lévy measure νx = ν.

In light of the Lévy-Khintchine formula, this means that the notion of a Lévy process and a hyperﬁnite Lévy process are essentially equivalent. In the proof of the subsequent result, the hyperﬁnite Lévy-Khintchine formula of Theorem 5.4.1 will play a crucial rôle.

Theorem 5.5.1. [263, Theorem 9.1] Given any generating triplet (γ,C,ν), there is a hyperﬁnite Lévy process X such that γ, C, ν are the drift vector, covariance matrix and Lévy measure of X’s standard part x =st(X) (cf. (5.1.1)), respectively.

Proof. It is enough to merely construct a hyperﬁnite Lévy process whose standard part has covariance matrix C and Lévy measure ν. We will pay no attention to the drift part, since after having constructed X with a ﬁnite drift ∗ d vector γ0 ∈ Fin( R ), we can always translate the set of increments of X by (γ − γ0)Δt to ensure that st(X) has drift vector γ.

◦ iy·X i y·x◦ Furthermore, we observe that by Lemma 5.3.3, E[e t ]=E[e t ] for all finite y ∈ Fin(∗Rd) and finite t ∈ T . Therefore and owing to the fact that we may ignore the drift part (as remarked previously), comparing the standard Lévy-Khintchine formula (Theorem 5.1.1) with the hyperfinite Lévy-Khintchine formula (Theorem 5.4.1) yields that for any hyperfinite Lévy ◦ process X, CX≤η is the covariance matrix of st(X) for any splitting infinitesimal. If we use Lemma 5.4.3 in addition, we see that νx will always be the Lévy measure of st(X).

So, we are left with the task of constructing a hyperﬁnite Lévy process X such that:

1. CX≤η ≈ C for some splitting inﬁnitesimal η,and −1 d 2. limRε↓0 L(ˆν)[st (D) \ Bε(0)] = ν(D) for all compact cubes D in R (cf. Remark 5.4.2).

First, we shall construct the part of X corresponding to ν (also known as the internal jump part). That is, we will look for an internal hyperfinite set ∗ d A 1 ⊂ R ( a)a∈A ∈ (0 1) 1 a 1 A and a family p 1 , with a∈A1 p < For this ∗ d 1 purpose, set BN = {y ∈ R | N ≤|y|≤N} and note that for all N ∈ N, ∗ ν[BN ]=ν[BN (0) \ B1/N (0)] is finite as ν is a Lévy measure. Since |y|≤N 5.5 Representation Theorem for Lévy Processes 233 ∗ for all y ∈ BN , this also implies the finiteness of |y| ν(dy) for all N ∈ N. BN Therefore, the set √ ∗ 1 ∗ N ≥ n, Δt · |y| ν(dy) ≤ , = ∈ N BN √ N An N ∗ 1 1 Δt · ν [BN ] ≤ N , Δt · N ≤ N is not only internal, but also nonempty, for all n. Furthermore, the sequence (An)n∈N is decreasing. The ℵ1-saturation of our nonstandard universe now ∗ yields that n∈N An = ∅ whence we have found an N ∈ N \ N such that √ √ ∗ ∗ Δt · |y| ν(dy) ≈ Δt · ν [BN ] ≈ Δt · N ≈ 0. (5.5.1) BN

Putting η =1/N, we will show later on that η is a splitting inﬁnitesimal. First, we have to construct the set A1, and for this sake we choose an internal lattice L of the d shape ζZ with inﬁnitesimal spacing ζ andinsuchawaythatL ∩ BN ⊂ Bη(0) (rather than merely ⊂ Bη(0)). Furthermore, we denote by ρ = ρL d the corresponding rounding operation ρL : R → L,thatis

⎛ ⎞d ⎜ ⎟ ⎜ ε ∗ ⎟ ρL : y → ⎜max ζZ ∩ yi + + R≤0 ⎟ . ⎝ 2 ⎠ =(ζZ) ≤yi+ε/2 i=1

∗ Then, all preimages of singletons in L under ρL are -Borel measurable, and so are hence all preimages of hyperﬁnite subsets of L under ρL.

Using this and the fact that L ∩ [−N,N]d is hyperﬁnite, we can deﬁne an internal measure νˆ on L via ∗ −1 νˆ : B → ν ρL (B) ∩ BN , and we may set A1 = {a ∈ L | νˆ{a} > 0} .

Also note that ∗ νˆ [A1]= νˆ{a} = νˆ{a} = νˆ{a} =ˆν [L]= ν [BN ]

a∈A1 a∈L a∈L νˆ{a}>0 234 5 Hyperﬁnite Lévy Processes and Applications is ﬁnite. Therefore, if we set

∀a ∈ A1 pa = Δtνˆ{a}, we get pa = Δt · νˆ [A1] ≈ 0.

a∈A1 =1− a ≈ 1 The complementary probability q a∈A1 p will be the weight that we will assign to the part of X that corresponds to C (also referred to as the internal diﬀusion part of X). We shall use the abbreviation 1 δ = apa Δt a∈A1 |a|≤1 in the following.

1/2 d×d 1/2 t 1/2 In order to deﬁne A2,chooseC ∈ R such that C = C · C t d×d ( D denoting the transpose of any D ∈ R ), and let {e1,...,ed} be the standard basis of Rd. We will set 1/2 Δt Δt ∀k ∈{1,...,d} a+k = C ek d − δ q q 1/2 Δt Δt ∀k ∈{1,...,d} a−k = −C ek d − δ q q

A2 = {a+1,a−1,...,a+d,a−d} q ∀a ∈ A2 pa = 2d A = A1 ∪ A2

Thus, A2 only contains 2d points, viz. just the a±k, k ∈{1,...,d},whichare 1 all assigned equal weight ≈ 2d . Note that by construction a∈A pa = a∈A pa + a∈A pa =(1− q)+ q 1 2 2d 2d =1, so we can indeed define a hyperfinite random walk with increments A = A1 ∪ A2 and transition probabilities (pa)a∈A. For the rest of this proof, X will always denote this hyperfinite random walk. Δt 1 As we shall see in a minute, subtracting δ q = q a∈A1 apa in the def- |a|≤1 inition of the a±k will ensure that the first condition of Theorem 5.2.2 is satisfied, and if we can show that the second and third conditions also hold for X, the said theorem may be applied to prove X to be a hyperfinite Lévy process (rather than merely a hyperfinite random walk). 5.5 Representation Theorem for Lévy Processes 235

|ρL(bi)| By the choice of L and ρL, one has |bi|≥ 2 for all i ∈{1,...,d} and ∗ d ∗ d b ∈ R . This implies |b|≥|ρL(b)|/2 for arbitrary b ∈ R . On the other hand, Rd \{0} | | ( ) the facts that ν is a Radon measure on and BM (0)\Bε(0) y ν dy < +∞ for all M,ε ∈ R>0 mean that |y| ν(dy) must be a Radon measure on the Polish space BM (0) \ Bε(0) for all M,ε ∈ R>0. Therefore, |y| ν(dy) is also outer-regular (cf. e.g. [76, Korollar 26.4]). This can be used, together with the nonstandard characterization of limits to get |y| ∗ν(dy) ≈ B1+ζ (0)\B1/N (0) 2 |y| ∗ν(dy).So, B1(0)\B1/N (0) 1 |a|pa = |a| νˆ(da) Δt B (0)\B (0) a∈A1 1 1/N |a|≤1 = |ρ(y)| ∗ν(dy) ρ−1(B (0))\ρ−1(B (0)) 1 1/N ≤ 2 |y| ∗ν(dy) ≈ 2 |y| ∗ν(dy). B1+ζ (0)\B1/N (0) B1(0)\B1/N (0)

Therefore and in light of (5.5.1) as well as the deﬁnition of δ, we may conclude that √ |δ| Δt ≤ 2 |y| ∗ν(dy) ≈ 0. (5.5.2) B1(0)\B1/N (0)

Hence all elements of A2 are inﬁnitely small.

1 a = − Combining this with the fact that Δt a∈A2 ap δ by construction, we arrive at 1 1 1 1 apa = apa + apa + apa Δt Δt Δt Δt a∈A a∈A1 a∈A2 a∈A1 |a|≤k |a|≤1 1<|a|≤k =−δ =δ = y νˆ(dy) (5.5.3) Bk(0)\B1(0) for all noninﬁnitesimal k. But, if we exploit once again that |b|≥|ρL(b)|/2 ∗ d ∗ for all b ∈ R and that νˆ is the image of ν under ρ when restricted to BN , we obtain

a νˆ(da) ≤ |a| νˆ(da) ≤ 2|y| ∗ν(dy) B (0)\B (0) B (0)\B (0) B (0)\B (0) k 1 k 1 k+ζ 1−ζ ≤ 2|y| ∗ν(dy) B2k (0)\B1/2(0) 236 5 Hyperfinite Lévy Processes and Applications which is finite for all finite and noninfinitesimal k as ν is a Lévy measure. 1 Thereby (5.5.3) actually implies that a∈A apa is finite for all finite Δt |a|≤k and noninfinitesimal k. Hence we have proven that the first condition of Theorem 5.2.2 holds for X.

In order to prove the second one, we employ the triangle inequality as well as (5.5.2) and the fact that q ≈ 1 to see that √ 1/2 dΔt δΔt ∀a ∈ A2 |a|≤ d max C · + i,j i,j q q

√ 1/2 ≈ d max C · Δt ≈ 0, (5.5.4) i,j i,j 2 2 a ≤ 1 | | a ≈ hence (exploiting that A is finite and a∈A2 p ), we get a∈A2 a p 0. Therefore we have for all k ∈ R 1 2 1 2 2 2 ∗ |a| pa ≈ |a| pa ≤ |a| νˆ(da) ≤ |y| ν(dy) Δt Δt B (0) B (0) a∈A a∈A1 k k+1 |a|≤k |a|≤k = |y|2 ν(dy) Bk+1(0) which is finite since ν is a Lévy measure. And because the expression on the 1 2 left is increasing in k, the finiteness of Δt |a|≤k |a| pa also follows for all finite k ∈ ∗R.

The third condition of Theorem 5.2.2 also holds for X: For all ﬁnite and ∗ noninﬁnitesimal k ∈ R>0, ( ) ( ) ( ) 1 ∗ pa =ˆν Bk(0) ≤ ν Bk−ζ (0) ≤ ν B◦2k(0) , Δt a∈A |a|>k which converges to zero as ◦k →∞since ν is a Radon (in particular, inner- d regular) measure on R \{0} satisfying ν[B1(0)] < +∞.

So we have veriﬁed all three conditions stated in Theorem 5.2.2, hence X is a hyperﬁnite Lévy process.

Moreover, in a similar way we can prove η to be a splitting inﬁnitesimal: 1 2 2 2 ∗ |a |pa ≤ |a| νˆ(da) ≤ |y| ν(dy) Δt a∈A Bc(0) B2c(0) |a|≤c = |y|2 ν(dy) B2c(0) 5.5 Representation Theorem for Lévy Processes 237 which converges to zero as c ↓ 0,since |y|2 ν(dy) < +∞ implies that B1(0) 2 |y| ν(dy) is a Radon measure (even a ﬁnite one) on the Polish space B1(0) ⊂ Rd and therefore |y|2ν(dy) is outer-regular (cf. e.g. [76, Korollar 26.4]).

What remains to prove is that CX≤η ≈ C and that νx[D]= −1 d limε↓0 L(ˆν)[st (D) \ Bε(0)] equals ν[D] for all compact cubes D in R . d −1 d ∗ 1 1 For any such cube D = i=1[ai,bi], st (D)= n∈N i=1 (ai− n ,bi+ n ) and one also has, since the internal measure νˆ is the image of ∗ν under ρ when restricted to BN , & ' d ∗ ∗ 1 1 ∗ ν ai − ,bi + \ B2ε(0) n +1 n +1 i=1& ' d ∗ 1 1 ∗ ≤ νˆ ai − ,bi + \ Bε(0) i=1 n n & ' d 1 1 ≤ ∗ ∗ − + \ ∗ (0) ν ai − 1,bi − 1 Bε/2 , i=1 n n that is & ' d 1 1 ν ai − ,bi + \ B2ε(0) n +1 n +1 i=1& ' d ∗ 1 1 ∗ ≤ νˆ ai − ,bi + \ Bε(0) i=1 n n & ' d 1 1 ≤ − + \ (0) ν ai − 1,bi − 1 Bε/2 i=1 n n for all n ≥ 2. Hence & ' d 1 1 inf ν ai − ,bi + \ B2ε(0) n∈N n n i=1 & ' d ◦ ∗ 1 1 ∗ ≤ inf νˆ ai − ,bi + \ Bε(0) n∈N i=1 n n & ' d 1 1 ≤ inf ν ai − ,bi + \ Bε/2(0) . n∈N i=1 n n

But by the deﬁnition of the Loeb measure and the outer regularity of the Radon measure ν on the Polish space Bε(0) (cf. again [76, Korollar 26.4]), this can be written as 238 5 Hyperﬁnite Lévy Processes and Applications & ' & ' d " d ∗ 1 1 ∗ ν [ai,bi] \ B2ε ≤ L (ˆν) ai − ,bi + \ Bε(0) i=1 n∈N i=1 n n & ' d ≤ ν [ai,bi] \ Bε/2(0) , i=1 hence −1 ∗ ν [D \ B2ε(0)] ≤ L (ˆν) st (D) \ Bε(0) ≤ ν D \ Bε/2(0) and therefore −1 ∗ lim L (ˆν) st (D) \ Bε(0) = lim ν [D \ Bε(0)] . ε↓0 ε↓0

If 0 ∈ D, then the compactness of D entails that we will find some ε ∈ R>0 −1 ∗ −1 such that D \ Bε(0) = D and st (D) \ Bε(0) = st (D), which readily −1 yields limε↓0 L(ˆν)[st (D) \ Bε(0)] = ν[D].Incase0 ∈ D and ν has finite mass, then ν can be extended to a Radon measure on Rd, which again will be outer-regular and therefore limε↓0 ν[D \ Bε(0)] = ν[D].If0 ∈ D and ν has infinite mass, then both limε↓0 ν[D \ Bε(0)] and ν[D] will be infinite, hence −1 also limε↓0 ν[D\Bε(0)] = ν[D]. Thus, even when 0 ∈ D, limε↓0 L(ˆν)[st (D)\ ∗ Bε(0)] = limε↓0 ν[D \ Bε(0)] = ν[D] which completes the proof of νx = ν.

The last step is to convince ourselves that CX≤η ≈ C. Recall that L was chosen such that L ∩ BN ⊂ Bη(0), hence |a| >η√ for all a ∈ A1.On the other hand, by virtue of relation√ (5.5.1), we have Δt/η ≈ 0, and due | | ∈ to estimate√ (5.5.4), also that a / Δt is ﬁnite for all a A2. Therefore, |a| | | = √ Δt ≈ 0 ∈ ⊂ (0) a /η Δt η for all a A2,inparticularA2 Bη . Summarizing this, A ∩ Bη(0) = A2 and therefore for all i, j ∈{1,...,d},

1 1 (C ) = a a p = a a p X≤η i,j Δt i j a Δt i j a |a|≤η a∈A2 d 1 q 1/2 Δt Δt 1/2 Δt Δt = C d − δi C d − δj Δt 2d i,k q q j,k q q k=1 d 1 q 1/2 Δt Δt 1/2 Δt Δt + − C d − δi − C d − δj Δt 2d i,k q q j,k q q k=1 d 2 1 q 1/2 1/2 Δt (Δt) = C C 2d + δiδj · 2d Δt 2d i,k j,k q q2 k=1 d Δt 1/2 1/2 = δiδk + C C q i,k j,k k=1 =((C1/2)·t(C1/2)) i,k 5.6 Extensions and Applications 239

1/2 t 1/2 1/2 2 2 However,√ (C ) · (C )=C by deﬁnition of C ,and|δiδj| ≤|δ| Δt = (|δ| Δt)2 ≈ 0 by relation (5.5.2); furthermore q ≈ 1. Hence we conclude

∀i, j ∈{1,...,d} (CX≤η )i,j ≈ Ci,j

and we are done.

5.6 Hyperﬁnite Lévy Processes: Extensions and Applications

Theorem 5.5.1 has been extended substantially by Albeverio and Herzberg [14]: They prove that the jump part of a Lévy process is essentially a convolution of Poisson processes in the following sense:

Theorem 5.6.1. [14, Theorem 4.4] Given any generating triplet (γ,C,ν), there is a hyperﬁnite Lévy process X,withA1 ⊂ L for some lattice L such that 1. γ,C,ν are the drift vector, covariance matrix and Lévy measure of X’s standard part x =st(X) (cf. (5.1.1)). 2. νˆ = ∗α∈L\{0}(λαδα +(1− λα)δ0),wheretheset{λα | α ∈ L} can be derived from ν via ∗ −1 λα = ν ◦ (ρL) {α}.

This result has a ﬁnancial interpretation in that it allows to regard a one-dimensional Lévy market model as essentially a binomial model which is unique modulo the order in which the jumps occur. In this sense, there is a very weak notion of completeness (viz. completeness up to the order in which the jumps occur) with respect to which any one-dimensional Lévy market model can be considered internally complete [14, Sects. 5, 6]; generalizations to arbitrary ﬁnite dimensions are conceivable, using the results on dynamically complete markets of Anderson and Raimondo [64].

However, the order in which the jumps occur can be neglected, if A1,theset of jumps, is suﬃciently small in cardinality when compared to the inverse time scale and some regularity parameters of the Lévy measure ν. This is basically the reason why they have to use the following very technically looking bound on the cardinality of L ⊃ A1 to prove Theorem 5.6.1:

Theorem 5.6.2. [14, Theorem 4.7] Let (γ,C,ν) be a generating triplet and use the notation of the proof of Theorem 5.5.1. Then there is a hyperﬁnite d Lévy process X,withA1 ⊂ L \{0} for some lattice L = εZ ∩ BN such that 1. γ,C,ν are the drift vector, covariance matrix and Lévy measure of X’s standard part x =st(X) (cf. (5.1.1)). 240 5 Hyperﬁnite Lévy Processes and Applications

∗ 2. N ∈ N \ N0 is chosen so small that 2 4D0,N +4D0,N D2,N + ND0,N D3,N + D0,N D1 + D2,N D1 2d·N 2d ·Δt · 3 · ((D0,N + D2,N ) ∨ 1) ≈ 0

= ∗ [ (0) \ (0)] where D0,N ,...,D3,N are deﬁned as D0,N ν BN B1 , 2 ∗ D1 = |x| ν(dx), D2,N = ν[B1(0) \ B1/N (0)],andD3,N = B1(0) ∗ |x| ∗ν(dx). B1(0)\B1/N (0)

Proof. Most of the theorem is a restatement of Lindstrøm’s representation theorem, Theorem 5.5.1. The fact that we can choose N suﬃciently small follows from the ℵ1-saturation principle since for all ﬁnite N, the hyperreals D0,...,D3 are standard real numbers: Setting 2 2d·N 2d 4D0,N +4D0,N D2,N + ND0,N D3,N + f(N):=3 · D0,N D1 + D2,N D1

((D0,N + D2,N ) ∨ 1) ,

one gets that the set , - · ( ) ≤ 1 √ Δt f N √ n , An := N ≥ n 1 1 Δt · D3,N ≤ n , Δt · N ≤ n

is not only internal, but also nonempty, for all n.Giventhat (An)n∈N is also decreasing, ℵ1-saturation yields an element N ∈ n∈N An. By deﬁnition of 1 An, the reciprocal of this N, N , will on the one hand serve as the splitting inﬁnitesimal η which we introduced in the proof of Theorem 5.5.1.Onthe other hand, N is going to satisfy Δt · f(N) ≈ 0. This completes the proof of Theorem 5.6.2.

Now we can outline the proof of Theorem 5.6.1 as well:

Proof (Proof sketch for Theorem 5.6.1). Thanks to Lindstrøm’s representation theorem for standard Lévy processes [263, Theorem 9.1], there is a hyperfinite adapted probability space Ω on which one can define a hyperfinite Lévy process Y· =(Yt)t∈T whose standard part y· has drift vector γ, covariance matrix C and Lévy measure ν, respectively, and which takes values in some lattice L.

In the proof of Theorem 5.5.1, however, the choice of the lattice spacing has been left open. On the other hand, (the second part of) the previous Theorem 5.6.2 asserts that we may assume 5.6 Extensions and Applications 241 2 2d·N 2d 4D0,N +4D0,N D2,N + ND0,N D3,N + Δt · 3 ((D0,N + D2,N ) ∨ 1) D0,N D1 + D2,N D1 ≈ 0.

In light of the trivial estimate card L ≤ (2 · N/ε)d that holds for any lattice 1 of mesh size ε and radius N, we may simply put ε = N and thereby obtain an inﬁnitesimal spacing for L which ensures that * + card L 2 3 Δt 4C0 +4C0C2 + NC0C3 + C0C1 +4C2C1 ((C0 + C2) ∨ 1) ≈ 0 in the notation of Theorem 5.6.2.

One can prove (cf. [14, Theorems 3.4 and 4.3]) that for all hyperfinite pure- jump Lévy processes Y with infinitesimal generator LΔt : f → α∈L λα · card L 2 (f(· + α) − f) and which satisfy 3 Δt(4C0 +4C0C2 + NC0C3 + C0C1 + 4C2C1)((C0 +C2)∨1) ≈ 0, there is a hyperfinite Lévy process X with internal infinitesimal generator * * + + ∗α∈L λαδα + 1 − λα δ0 ∗ f − f KΔt : f → Δt which has the same standard part (pointwise for all smooth functions with compact support and all elements of L) as the internal infinitesimal generator LΔt.

However, one can also establish (cf. [14, Lemma 4.5]) that two hyperfinite pure-jump Lévy processes whose internal infinitesimal generators have the same standard part (pointwise for all smooth functions with compact support and all elements of L), will have standard parts with the same infinitesimal generator, thus standard parts with identical finite-dimensional distributions. Hence the standard part x· =st(X) coincides with the standard part y· = st(Y ). This brings the proof to a close.

As another application, Lindstrøm himself [264] has used hyperﬁnite Lévy processes to develop a theory of nonlinear stochastic integration with respect to Lévy processes and to study certain minimal martingale measure problems for stochastic integrals with respect to a Lévy process. Current research on hyperﬁnite Lévy processes includes a pathwise theory of stochastic integration with respect to (smooth functions of) Lévy processes with jump part of bounded variation [205] and applications in equilibrium asset pricing with stochasticity derived from Lévy processes [204].

Chapter 6 Epilogue: Genericity of Hyperﬁnite Loeb Path Spaces

An important part of this book has been concerned with the investigation of hyperfinite Markov chains X =(Xt)t∈T on an internal probability space Ω. If we confine ourselves to a finite time horizon – say, 1 – and if we assume Δt to be a reciprocal hypernatural number, then H =1/Δt ∈ ∗N \ N will be the number of possible transitions occurring between 0 ∈ T and 1 ∈ T . Because X was assumed to be a hyperfinite Markov chain starting at a given point x0 in the state space, the number of possible paths between 0 and 1 will be hyperfinite. Thus, provided we are only interested in studying events that lie in the filtration generated by X, we may assume Ω to be hyperfinite.

Already very early in the history of nonstandard probability theory, important Markov processes, such as the Brownian motion or the Poisson process, have been found to have simple internal analogues (by [61,267], respectively), and in Chap. 5, we have seen how to construct an internal analogue of an arbitrary Lévy process.

However, the analogy between the original Lévy process (on an arbitrary probability space) and the standard part of its internal analogue (on an internal probability pace) is limited to the equality of the corresponding Markovian semigroups, and hence the finite-dimensional distributions. It cannot be assumed a priori that the original process – defined on an arbitrary given probability space – and the standard part of its internal analogue on some hyperfinite probability space will be indistinguishable by propositions in a suitably formalized language of adapted probability theory with a finite time horizon.

We will see in this final chapter that this is indeed the case – even for arbitrary, not necessarily Markovian stochastic processes –, and that in the special case of Markov processes, on which this book is putting particular emphasis, indistinguishability with respect to the language of adapted probability theory with a finite time horizon is the same as equality of the respective processes’ finite-dimensional distributions.

S. Albeverio et al., Hyperﬁnite Dirichlet Forms and Stochastic Processes, 243 Lecture Notes of the Unione Matematica Italiana 10, DOI 10.1007/978-3-642-19659-1_6, c Springer-Verlag Berlin Heidelberg 2011 244 6 Genericity of Loeb Path Spaces

In this short chapter, we shall not give any proofs and formal references. Rather, we will largely follow Fajardo and Keisler’s comprehensive, however dense, monograph [156] and refer the interested reader to their work as well as to the original articles by Hoover and Keisler [210] and Keisler [238].

6.1 Adapted Probability Logic

Deﬁnition 6.1.1. Atriple(Ω,(Ft)t∈[0,1],P) consisting of a set Ω,aﬁltra- tion (Ft)t∈[0,1] over Ω and a probability measure P : F1 → [0, 1] is called an adapted probability space (for short: adapted space).

An adapted function can be viewed as a scheme to construct a real-valued random variable from a stochastic process on a given Polish space – provided that this random variable only depends on the process at ﬁnitely many times and that one conﬁnes oneself to the use of bounded continuous functions and conditional expectations.

Deﬁnition 6.1.2 (Adapted functions and their values). Let M be a Polish space. The set of adapted functions is deﬁned inductively as the smallest set of expressions f such: that

1. Base Step. For all n ∈ N and any n-tuple (t1,...,tn) of mutually distinct elements of [0, 1] and for all bounded continuous functions φ : M n → R,

φt1,...,tn is an adapted function. If (xt)t∈[0,1] is an M-valued stochastic

process, then the R-valued random variable φt1,...,tn (x·)=φ(xt1 ,...,xtn )

is the value of φt1,...,tn for x·. 2. Composition Step. Let m ∈ N, assume ψ : M m → R is bounded and continuous, and let f1,...,fm be adapted functions. Then ψ(f1,...,fm) is an adapted function as well, and if (xt)t∈[0,1] is an M-valued stochastic process, then the real-valued random variable ψ(f1,...,fm)(x·)= ψ(f1(x·),...,fm(x·)) is the value of ψ(f1,...,fm) for x·. 3. Conditional Expectation Step. Let t ∈ [0, 1] and let f be an adapted function. Then, E[f|t] also is an adapted function, and if (xt)t∈[0,1] is an M-valued stochastic process on an adapted probability space (Ω,(Ft)t∈[0,1],P), then the real-valued random variable E[f|t](x·)= EP [f(x·)|Ft] is the value of E[f|t] for x·.

This gives rise to the following notion for the equivalence of two processes:

Deﬁnition 6.1.3. Let M be a Polish space, let (xt)t∈[0,1] be an M-valued stochastic process on an adapted probability space (Ω,(Ft)t∈[0,1],P) and let (yt)t∈[0,1] be an M-valued stochastic process on an adapted probability space (Γ, (Gt)t∈[0,1],Q). The processes (xt)t∈[0,1] and (yt)t∈[0,1] shall be called adapted-equivalent (in symbols: x· ≡ y·) if and only if for all adapted functions f, one has EP [f(x·)] = EQ[f(y·)]. 6.2 Universality, Saturation, Homogeneity 245

This notion is at least as strong as equality of ﬁnite-dimensional distributions:

Lemma 6.1.1. Let (xt)t∈[0,1] be an M-valued stochastic process on an adapted probability space (Ω,(Ft)t∈[0,1],P) and let (yt)t∈[0,1] be an M-valued stochastic process on an adapted probability space (Γ, (Gt)t∈[0,1],Q).Ifx· ≡ y· 1 n ∈ [0 1] = and t ,...,t , are mutually distinct, then P(xt1 ,...,xtn ) Q(yt1 ,...,ytn ) as probability measures on M n. We shall see later on, in Theorem 6.4.1, that the converse holds for Markov processes (due to a theorem of [210]).

6.2 Universality, Saturation, Homogeneity

For the rest of this chapter, we assume all processes to have Polish spaces as state spaces.

First, the – fairly self-explanatory – concept of universality is introduced:

Deﬁnition 6.2.1. Let Ω =(Ω,(Ft)t∈[0,1],P) be an adapted probability space. The adapted space Ω is adapted-universal if and only if for all stochastic processes y· on any other adapted probability space, there is a stochastic process x· on (Ω,(Ft)t∈[0,1],P) such that x· ≡ y·.

For the next Deﬁnition, note that whenever M is a Polish space, so will be M 2 (and, in fact, any M n for n ∈ N). Hence, adapted functions (as well as the notion of adapted equivalence) are also deﬁned for M 2-valued stochastic processes.

If an adapted probability space Ω has the following property, many probabilistic arguments on any other adapted probability space can be translated into arguments on Ω:

Deﬁnition 6.2.2. Let Ω =(Ω,(Ft)t∈[0,1],P) be an adapted probability space. The adapted space Ω shall be called adapted-saturated if and only if for all Γ =(Γ, (Gt)t∈[0,1],Q) and for all stochastic processes x· on Ω and y·,z· on Γ satisfying x· ≡ y·, there exists another stochastic process w· on Ω such that (x·,w·) ≡ (y·,z·). An adapted probability space can be regarded as uniquely determined by the expected values of adapted functions whenever it satisﬁes the following condition:

Deﬁnition 6.2.3. If Ω =(Ω,(Ft)t∈[0,1],P) is an adapted probability space, then Ω shall be called adapted-homogeneous if and only if for all stochastic processes x·,y· on Ω satisfying x· ≡ y·, there exists a bijection h : Ω0 ↔ Ω1 246 6 Genericity of Loeb Path Spaces such that:

1. P [Ω0]=P [Ω1]=1, 2. for all t ∈ [0, 1], h[Ft]=Ft (where Ft denotes the completion of the σ-algebra Ft with respect to P ), 3. for all A ∈F1, P [h(A)] = P [A],and 4. for all t ∈ [0, 1], one has xt = yt ◦ hP-almost surely. (Such a map h is called adapted automorphism.)

Finally, these three features of an adapted probability space are not unrelated – saturation implies universality, and homogeneity together with universality implies saturation:

Theorem 6.2.1. Let Ω =(Ω,(Ft)t∈[0,1],P) be an adapted probability space. Then: • If Ω is both adapted-universal and adapted-homogeneous, then it is adapted-saturated. • If Ω is adapted-saturated, then it is adapted-universal.

6.3 Hyperﬁnite Adapted Spaces

Definition 6.3.1. A hyperfinite adapted probability space is a triple (Ω,(Ft)t∈[0,1],L(μ)) such that: T • Ω = Ω0 ,whereΩ0 is a hyperfinite set with at least two elements and 1 H!−1 ∗ T = {0, H! ,..., H! , 1} (for H ∈ N \ N)isthehyperfinite time line. • The external filtration (Ft)t∈[0,1] is derived from some internal filtration (As)s∈T over Ω via Ft = σ( ◦s=t As) for all t ∈ [0, 1],and Ω (∀r ≤ s ∀ω ∈ Ω ∀ω ∈ Ωω(r)=ω (r)) As = A ∈ 2 . ⇒ (ω ∈ A ⇔ ω ∈ A)

• L(μ) is the Loeb extension of the internal normalized counting measure μ : A →|A|/|Ω| on Ω.

Example 6.3.1. For instance, the probability space on which Anderson’s lift- T ing (Bt)t∈T of the Brownian motion [61] is defined, is Ω = {−1, 1} .Let (As)s∈T be the internal filtration generated by this lifting and derive an external filtration F· from A· in the manner of the previous Definition 6.3.1. Also, let μ be the joint internal distribution of all the Bt, t ∈ T –whichis a finitely additive probability measure on A1.Then(Ω,(Ft)t∈[0,1],L(μ)) is a hyperfinite adapted probability space. 6.4 Probability Logic and Markov Processes 247

Theorem 6.3.1. Every hyperﬁnite adapted probability space (Ω,(Ft)t∈[0,1], L(μ)) is adapted-universal, adapted-homogeneous, and thus also adapted- saturated. The witness for the adapted homogeneity can even be chosen as an internal bijection h : Ω ↔ Ω.

The special importance of hyperﬁnite adapted probability spaces is empha- sized by the fact that no Polish adapted probability space is universal, let alone saturated:

Theorem 6.3.2. Let Ω be a Polish space, let Q be the completion of a probability measure on the Borel σ-algebra of Ω,andlet(Ft)t∈[0,1] be a ﬁl- tration such that each element of Ft is Q-measurable, for all t ∈ [0, 1].Then (Ω,(Ft)t∈[0,1],Q) cannot be universal.

6.4 Adapted Probability Logic and Markov Processes

For Markov processes – and hardly any other stochastic processes have been treated in this book –, conditioning with respect to the information up to a certain time actually can be reduced to simply looking at the transition kernel at this given time, since Markov processes have no memory. In this sense, we should not be too surprised by the following result (the converse to Lemma 6.1.1):

Theorem 6.4.1. Any two Markov processes x· on Ω and y· on Γ are adapted-equivalent if and only if their ﬁnite-dimensional distributions are equal (i.e. they are equivalent in the sense of, e.g., [77]).

Summarizing this short chapter, we have seen that hyperfinite adapted probability spaces can be regarded as the generic choice of an adapted probability space, in a rigorous sense (referring to adapted universality, saturation and homogeneity). Furthermore, as Theorem 6.4.1 states, any two Markov processes x· and y· with the same finite-dimensional distributions cannot be discerned by adapted functions in expectation: There is no adapted function which would attain a different expected value when applied to x· instead of y·.

We close this book by addressing two possible foundational objections against nonstandard analysis. First, it has to be acknowledged that many foundational questions about nonstandard analysis (particularly with regard to its set-theoretic strength) can be resolved by adopting a syntactic approach to nonstandard analysis, as developed by Nelson [284] (Internal Set Theory, IST), Hrbáček [211] and Kanovei and Reeken [228] (Bounded Set Theory, BST). As part of this well-developed and still-growing literature, several authors have proposed nonstandard set theories, i.e. axiomatisations of 248 6 Genericity of Loeb Path Spaces

Robinson’s nonstandard analysis (and fragments thereof), and proven that they are conservative extensions of subsystems of ZFC.

On the other hand, a folklore objection to the application of nonstandard methods in analysis used to be the well-known fact that the existence of nonstandard enlargements is dependent on the choice of a non-principal ultrafilter and is thus somewhat “non-constructive”. (Indeed, the Ultrafilter Existence Theorem does not follow from Zermelo-Fraenkel set theory only.) However, this objection against the use of nonstandard analysis can be refuted in at least two ways. Firstly, the Ultrafilter Existence Theorem – which is equivalent to the Boolean Prime Ideal Theorem – is not as strong as the Axiom of Choice (cf. [200]aswellas[71]). Secondly, recent studies in model theory have revealed that there exist nonstandard universes which are definable through a formula of set theory: Kanovei and Shelah [230] were the first to prove the existence of a definable nonstandard model of the reals; Kanovei and Reeken [229] even established the existence of definable nonstandard set universes, and Herzberg [202, 203] showed that there are nonstandard universes in the usual sense (as enlargements of the superstructure over the reals). Admittedly, the Axiom of Choice is still employed to prove that the structures thus defined have the properties of nonstandard models – in particular, κ-saturation for uncountable cardinals κ. Nevertheless, we now know that nonstandard universes are far more tractable objects than popular opinion expected only one decade ago. Appendix

We summarize here some basic facts of nonstandard analysis. For details and proofs, see e.g. Albeverio et al. [25], Courant and Robbins [123], Courant et al. [124], Cutland [125], Dauben [130], Davis [135], Stroyan and Bayod [341], and Stroyan and Luxemburg [342].

A.1 General Topology

Let Y be a Hausdorff topological space. First of all, let us construct ∗Y .Let η be a finitely additive measure on the set N of positive integers such that (a) For all A ⊂ N, η(A) is defined and either 0 or 1. (b) η(N)=1, and η(A)=0for all finite A ⊂ N.

It is well-known that such a measure exists as a consequence of, e.g., the axiom of choice.

Given two sequences y = {yn}n∈N and z = {zn}n∈N, yn,zn ∈ Y , we deﬁne the following equivalence relation

y ∼ z if and only if η({n ∈ N | yn = zn})=1.

Having deﬁned this relation, we introduce

∗Y = Y N/ ∼ .

We call ∗Y the nonstandard version or extension of Y . In particular, we call ∗R = RN/ ∼ the nonstandard reals or hyperreals. In the same manner, we can deﬁne ∗N, etc.

Fixed an element y ∈ Y ,themonad of y is the subset of ∗Y deﬁned by

μ(y)=∩{∗O | y ∈ O and O is open}.

S. Albeverio et al., Hyperﬁnite Dirichlet Forms and Stochastic Processes, 249 Lecture Notes of the Unione Matematica Italiana 10, DOI 10.1007/978-3-642-19659-1, c Springer-Verlag Berlin Heidelberg 2011 250 Appendix

An element a ∈ ∗Y is a nearstandard point if it belongs to μ(y) for some y ∈ Y . The set of all nearstandard points is denoted by Ns(∗Y ).SinceY is Hausdorﬀ, each element a ∈ Ns(∗Y ) is nearstandard to exactly one element y of Y. Let us call y the standard part of point a and denote it by ◦a or st(a). Moreover, we have the following characterizations of open, closed and compact sets. If A is a subset of ∗Y , the standard part of set A is deﬁned by

◦A =st(A)={y ∈ Y |∃s ∈ A(st(s)=y)}.

Proposition A.1.1. Let A be subset of Y .Then (i) A is open if and only if st−1(A) ⊂ ∗A. (ii) A is closed if and only if ∗A ∩ Ns(∗Y ) ⊂ st−1(A). (iii) A is compact if and only if ∗A ⊂ st−1(A).

A subset A of ∗Y is called S-dense in ∗Y if for all y ∈ Y ,thereisan element a ∈ A such that a ∈ μ(y). A subset A of ∗Y is called S-rich in ∗Y if it is hyperﬁnite and S-dense in ∗Y .

A.2 Structure of ∗R

N For two sequences a = {an | n ∈ N} and b = {bn | n ∈ N} in R , let us deﬁne addition and multiplication operations by

a + b = {an + bn | n ∈ N} and ab = {anbn | n ∈ N}.

The zero for addition, denoted by 0, is the sequence with only zeros, while the unit of multiplication, denoted by 1, is the sequence with only ones.

N For any a = {an | n ∈ N} in R ,wedenotebya its equivalence class with respect to ∼.Itiseasytoverify

a + b = a + b, ab = ab.

−1 −1 −1 If a =0we can define the inverse a by (a )n =(an) whenever ∗ ∗ an =0. The above definitions make R into a field of numbers. R can be linearly ordered by defining

a < b if and only if η({n ∈ N | an

We can consider R as embedded in ∗R by identifying any r ∈ R with ∗r ∈ ∗R wherealltheelementsequalr. From now on, we will write r for the element ∗r ∈ ∗R. A.3 Internal Sets and Saturation 251

Let a ∈ ∗R,wecalla inﬁnitesimal if and only if for all r ∈ R,r >0

−r

We call a ∈ ∗R inﬁnite if and only if for all r ∈ R,r >0

|a| >r,

∗ where |a| = {|an|}n∈N for a = {an}n∈N, and a ∈ R is called ﬁnite if and only if there exists an r ∈ R,r >0 such that

|a|

Proposition A.2.1. Any ﬁnite a ∈ ∗R can be written uniquely as

a = r + ε (A.2.1)

with r ∈ R and ε inﬁnitesimal.

We call r in the relation (A.2.1)thestandard part of a and denote it by st(a) or ◦a. Conversely, for each r ∈ R,thesetofalla ∈ ∗R such that a ≈ r is called the monad of r.

A.3 Internal Sets and Saturation

∗ A sequence {An}n∈N of subsets of R deﬁnes a subset {An}n∈N of R by

{xn}n∈N∈{An}n∈Nif and only if η({n | xn ∈ An})=1.

A subset of ∗R which can be obtained in this way is called internal. A subset of ∗R which is not internal is called external.

Proposition A.3.1. Let A be an internal subset of ∗R. (i) If A contains arbitrarily large elements, then A contains an infinite element. (ii) If A contains arbitrarily small positive infinite elements, then A contains a finite element.

i Proposition A.3.2. Let {A }i∈N be a sequence of internal sets such that i i ∩i≤I A = ∅ for all I ∈ N, then ∩i∈NA = ∅.

Proposition A.3.2 is known as the Principle of Countable Saturation or ℵ1- Saturation. (It is possible to construct nonstandard universes which satisfy even higher saturation principles.) 252 Appendix

Proposition A.3.3. Let {An}n∈N be a sequence of internal sets. The union ∪n∈NAn is internal if and only if it equals ∪n≤N An for some N ∈ N.

A.4 Loeb Measure

For any set S, we will write P(S) for the set of all subsets of S. Let us deﬁne

a sequence {Vn(S)}n∈N0 of sets recursively by

V0(S)=S, Vn+1(S)=Vn(S) ∪P(Vn(S)).

The superstructure over S is the union V (S)=∪n∈N0 Vn(S). In the same way as in A.1, we can construct ∗S by

∗S = SN/ ∼ .

Let Ω be an internal set in some superstructure V (∗S).Aninternal algebra on Ω is an internal set F of subsets of Ω which contains ∅ and Ω, and is closed under complements and finite unions. Since F is internal, it automatically satisfies a stronger property: if F1 is a hyperfinite subset of F, the union ∪{A | A ∈F1} is an element of F.

∗ ∗ Let R+ = R+ ∪{0, ∞} be the set of extended, nonnegative hyperreals. An internal, ﬁnitely additive measure on the internal algebra F is an internal ∗ function μ : F−→ R+ such that μ(∅)=0and

μ(A ∪ B)=μ(A)+μ(B)

for all disjoint A, B ∈F.Sinceμ is internal, the additivity extends to the following hyperﬁnite unions:

μ(∪A∈F1 A)= μ(A)

A∈F1

for all disjoint, hyperﬁnite subsets F1 of F.

Definition A.4.1. (i) A subset B of Ω is μ-approximately if for each ε ∈ R+, there are sets A, C ∈F such that A ⊂ B ⊂ C and μ(C) − μ(A) <ε. (ii) The Loeb algebra L(F) consists of those B ⊂ Ω such that B ∩ F is μ-approximable for all F ∈F with finite μ-measure. (iii) The Loeb measure of μ is the map L(μ):L(F) −→ R+ defined by

L(μ)(B)=inf{◦μ(C) | C ∈F and C ⊃ B}.

Proposition A.4.1. (Ω,L(F),L(μ)) is a complete measure space. A.5 Linear Spaces 253 A.5 Linear Spaces

Let (E,|| · ||) be a linear normal space. An element x ∈ ∗E is called finite if ||x|| is a finite hyperreal. Let us denote by Fin(∗E) the set of all finite elements of ∗E.Wecallx ∈ ∗E infinitesimal if ||x|| ≈ 0. We write x ≈ y if ||x − y|| ≈ 0.Anelementx ∈ ∗E is called nearstandard if ||x − y|| ≈ 0 for ∗ some y ∈ E. An element x ∈ E is called pre-nearstandard if for all ε ∈ R+ there is some y ∈ E such that ||x − y|| <ε.WedenotebyPns(∗E) the set of all pre-nearstandard points of ∗E.

Proposition A.5.1. The space Pns(∗E)/ ≈ is the completion of the given space E.

Proposition A.5.2. Let (F, || · ||) be an internal normed linear space. Then Fin(F )/ ∼ is a Banach space with the norm |||◦x||| = ◦||x||, where ◦x denotes the equivalence class of x ∈ Fin(F ).

For N ∈ ∗N, an internal space F is called N-dimensional if there is an internal set {e1,e2, ··· ,eN } such that each element x ∈ F can be expressed N by x = i=1 xiei uniquely.

Proposition A.5.3. Let E be an inﬁnite-dimensional linear space over R. Then, there exists a hyperﬁnite-dimensional space X over ∗R such that E ⊂ X ⊂ ∗E.

Historical Notes, Bibliographical Complements

Chapter 1

The theory of Dirichlet forms has its origins in seminal work by Beurling and Deny [87,88]. It relies essentially on the use of Hilbert space methods in potential theory, which in turn originate from “energy methods” and “variational methods”. Such methods of potential theory were developed particularly during the period 1935–1950 by analysts such as C.J. De la Vallée Poussin, M. Riesz, O. Frostman and H. Cartan, see, e.g., Cartan [106]. The essential new idea in Beurling and Deny’s fundamental work consists in the systematic use of the Dirichlet norm and the corresponding contraction properties of Dirichlet forms.

In the theory of Dirichlet forms, the case of discreteness (in time and/or space) has played, at least in the beginning, an important role, as it facilitated the description of the underlying ideas and methods without having to dwell on the technical problems encountered in implementing the basic ideas of the theory in the continuum case.

Discrete-time potential theory was developed further by J. Deny (e.g. [142], see also [143]), and J.L. Doob [145,146], see also, e.g., Revuz [306], Williams [360,361], Norris [291], Dellacherie and Meyer [138]. The hyperﬁnite approach to Dirichlet forms originated in work by Albeverio, Fenstad and Høegh-Krohn [10], as well as Albeverio, Fenstad, Høegh-Krohn and Lindstrøm [25]. One of its aims was to preserve as much as possible the spirit of the discrete-time and discrete-space approach, by considering a hyperﬁnite model both in time and space which by its very essence is discrete but at the same time yields an alternative view of continuum realizations.

Symmetric, nonnegative (or lower semi-bounded) quadratic forms and their association with nonnegative (or lower semi-bounded) self-adjoint operators is a well-developed topic of analysis, occurring, e.g. in the relation with the calculus of variations or problems of spectral theory and quantum mechanics, see e.g. Rellich [303], Reed and Simon [300–302], Faris [168],

255 256 Historical Notes, Bibliographical Complements

Weidmann [356–358], Bratteli and Robinson [101], Kato [235], Arendt and Thomaschewski [67], Zhang [365], Takeda [348], Lenz, Stollmann and Veselić [260].

Forms which are unbounded from below are much less studied, see, however, e.g., McIntosh [279, 280] or Berrahmoune [85].

Nonsymmetric (lower semi-bounded) bilinear forms occur in the theory of dissipative processes, see, e.g., Pazy [299] and Arendt, Batty, Hieber and Neubrander [66], Kuwae [252], Hu, Ma and Sun [214].

The relation with corresponding operators is studied, e.g. in Albeverio and Ma [41, 42], Albeverio, Ma and Röckner [43–45], and Pazy [299].

For applications of nonsymmetric Dirichlet forms, see, e.g., Hu, Ma and Sun [213, 214].

For questions of self-adjoint (maximal dissipative, respectively) extensions, see, e.g., the references in Remark 1.2.1.

For the origins of the theory of nonsymmetric Dirichlet forms, see Albeverio [2].

For applications of the theory of Dirichlet forms to manifolds, see, e.g., Ouhabaz [297], Davies [131, 132], Cycon, Froese, Kirsch and Simon [127], Biroli and Vernole [93], and Bonciocat [97].

For extensions of the theory of Dirichlet forms to the non-commutative case, see Albeverio and Høegh-Krohn [18], Davies and Lindsay [133], Cipriani [120] and for the supersymmetric case Albeverio and Kondratiev [33].

For the case of p-adic state spaces, see Albeverio, Karwowski and Yasuda [29], Albeverio, Karwowski and Zhao [30], Kochubei [243, 244], Kaneko and Yasuda [227], Kaneko [226], and for more general trees, see Evans [153]or Albeverio and Karwoski [28].

For the case of forms related to jump processes, see, e.g., Sato [323], Apple- baum [65], Fukushima and Tanaka [186], Uemura [351], Barlow, Bass, Chen and Kassmann [72], Foondun [173], and Chen and Kumagai [115].

For the study of Dirichlet forms on graphs, see, e.g., Bonciocat [97], vonBelowandLubary[79, 80], von Below and Nicaise [81], Lumer [269], and Kant, Klauss, Voigt and Weber [231]. For applications of Dirichlet forms to (quantum) graphs and discrete spaces, see, e.g., Bonciocat [97].

For the problem of tubular neighbourhoods of graphs and diﬀusion operators associated to them and their graph limits, see Albeverio, Cacciapuoti Historical Notes, Bibliographical Complements 257 and Finco [7] and also the appendix by P. Exner in Albeverio, Gesztesy, Høegh-Krohn, and Holden [12].

For Dirichlet forms in inﬁnite-dimensional analysis, see, e.g., Albeverio [2], Chen, Ma, and Röckner [116], Eberle [149], Liskevich and Röckner [265], Da Prato [128], Huang and Yan [215], Wang and Wu [354,355], Ma and Röck- ner [270], Denis, Grorud and Pontier [141], Albeverio, Kondratiev, Kozitsky, and Röckner [31] (and Carmona and Tehranchi [104] for applications to interest-rate models in mathematical ﬁnance).

For applications of Dirichlet forms to Schrödinger operators, see, e.g., Albeverio [1], Albeverio, Høegh-Krohn, and Streit [26], Albeverio and Ma [40, 41], Demuth and Krishna [140], Wu [363], Robinson [312], Davies and Simon [134], Kuwae [250], and Laptev and Weidl [256] (see also, e.g., Exner [154]andDavies[132] for relations to dissipative systems).

For relations to the theory of metric measure spaces and spectral theory, see Bonciocat [97], Sturm [343], Paulik [298], Kuwae and Shioya [253, 254], and Grigoryan and Hu [195].

For convergence of families of Dirichlet forms, see, e.g., Albeverio, Kusuoka and Streit [36], Stollmann [339], Kuwae and Shioya [253, 254], Mosco [283], Ben Amor and Brasche [82], Kolesnikov [245], Biroli [91], and Biroli and Tchou [92].

For the construction of Markov processes from a (nonsymmetric) Lp- semigroup, see, e.g., Jacob [220].

For relations between Dirichlet forms, semigroups, and processes, see, e.g., Feller [170], Ma and Röckner [270], Fuhrman [174], Albeverio [2], Taira [344], Fukushima [179, 180], and Chen, Fitzsimmons, Kuwae, and Zhang [111], LeJan [258], and Ôkura [292] (see also Oshima [294]) and Stroock [340].

For the relation of Dirichlet forms with general potential theory, see Beznea and Boboc [89], Beznea, Boboc and Röckner [90], and Bliedtner and Hansen [95]. In particular, quasi-regular semi-Dirichlet forms and their relations with right processes are studied in the former paper.

For relations to non-equilibrium particle systems, random walks on fractals and statistical physics, see, e.g., Chen [109].

For relations of discrete Dirichlet forms to fractals, see, e.g., Chen [109], Kigami [239], Hino [209],andHuandZähle[212] (Kigami [239]alsotreats approximations of Dirichlet forms).

Higher-order operators in relation to Dirichlet forms are discussed, e.g., in Noll [290]. 258 Historical Notes, Bibliographical Complements

For the theory of generalized Dirichlet forms, see, e.g., Stannat [336,337], Trutnau [349, 350], and Conrad and Grothaus [122].

For an analytic treatment of generators of Markov semigroups with Hölder- continuous coeﬃcients, see Lorenzi and Bertoldi [268].

For nonlinear Dirichlet problems and their relations to the theory of harmonic mappings, see, e.g., Hesse [207] and Hesse, Rumpf and Sturm [208].

Canonical diffusion processes on metric measure spaces have been constructed by von Renesse [304]andvonRenesseandSturm[305]. For relations of Dirichlet forms with gradient-flows in metric spaces and the theory of trans- port in spaces of probability measures, see, e.g., Jordan, Kinderlehrer, and Otto [224], Ambrosio, Gigli, and Savaré [58]. For Dirichlet forms associated with log-concave measures and Fokker-Planck equations, see Albeverio and Kusuoka [34], Albeverio, Kusuoka, and Röckner [35], Ambrosio, Savaré, and Zambotti [59]. For recent work on Wasserstein diffusions, see also Andres and von Renesse [60].

Gauging of Dirichlet forms is discussed in von Renesse [304], Takeda [347], Zhao [366], and Chen and Shiozawa [117].

Applications of Dirichlet forms to statistics (sampling perfect matchings of a graph) are given in Jerrum [223].

Applications to boundary value problems can be found in Bass and Hsu [74], Krall [246], Bass and Kassmann [75], Fang, Fukushima, and Ying [167], Chen, Fukushima, and Ying [114], Chen and Fukushima [110], and applications to semilinear diﬀerential equations have been studied by Stollmann and [339], Vogt and Voigt [352], Chen and Zhang [118], and Grossinho and Tersian [197].

Sobolev-type inequalities for Dirichlet forms and corresponding hypercontractivity estimates are discussed, e.g., in Gross [196], Fabes, Fukushima, Gross, Kenig, Röckner, and Stroock [155], Wang [353], Mendez-Hernandez [281], Saloﬀ-Coste [322], Kassmann [233], Del Moral, Ledoux, and Miclo [137], Cattiaux [107],DaiPra,andPosta[129], and Cattiaux, Gentil, and Guillin [108], Grunewald, Otto, Villani, and Westdickenberg [199], Otto and Reznikoﬀ [296] (see also, e.g., Gestesy, Holden, Jost, Paycha, and Röckner and Scarlatti [182, 183]), Bakry [69], Bakry, Concordet, and Ledoux [70], Albeverio, Kozitsky, Kondratiev, and Röckner [30].

For Dirichlet forms related to the Lévy Laplacian, see M.N. Feller [169]as well as Albeverio, Belopolskaya, and M.N. Feller [3,4] and references therein.

For problems of pseudodiﬀerential operators and Dirichlet forms, see, e.g., Bass and Kassmann [75], Schilling and Uemura [324], or Kassmann [234]. Historical Notes, Bibliographical Complements 259

For random Dirichlet forms, see, e.g., Fukushima, Nakao, and Takeda [182], Albeverio and Bernabei [5], and Rhodes [308].

Chapter 2

The relation between probabilistic potential theory and the theory of Dirich- let forms goes back to the origins of the theory of Dirichlet forms, see the notes in Chap. 1.

Fukushima’s decomposition theorem has been extended to the inﬁnite- dimensional setting in Albeverio, Ma, and Röckner [38]andinthebookby Ma and Röckner [270].

For the theory of Feynman-Kac functionals, see, e.g., Albeverio and Ma [39–42], Albeverio, Blanchard, and Ma [6], Brasche [100], De Leva, Kim, and Kuwae [136], Eberle and Marinelli [150],andDemuthandVanCasteren[139], as well as Albeverio and Fan [8].

For the theory of subordination, see, e.g., Albeverio and Rüdiger [51, 53], Jacob [220], Jacob and Moroz [221], Chen, Fukushima, and Ying [113], Fitzsimmons [171], Fukushima and Uemura [187], Fukushima, He, and Ying [181], and Kim, Song, and Vondraček [242]. For a recent application, see Albeverio and Gordina [13].

Chapter 3

The standard representation theory has its origins in Albeverio, Fenstad, Høegh-Krohn and Lindstrøm [25]. It uses heavily results of Lindstrøm [261] and Fukushima [175],

Chapter 4

The association of “good” Markov processes to Dirichlet forms goes back to work by Silverstein [333, 334], Fukushima [175], Fukushima, Ôshima and Takeda [183] in the symmetric case, and to Carrillo-Menendez [105], and Albeverio, Ma, and Röckner [44] in the nonsymmetric case. The nonstandard analytic framework is treated in Albeverio, Fenstad, Høegh-Krohn, and Lindstrøm [25], and Fan [165]. See also Albeverio [2] and references therein. Quasi-regularity was found and elaborated in Albeverio and Ma [41]andMa and Röckner [270]. 260 Historical Notes, Bibliographical Complements Chapter 5

The theory of Lévy processes goes back to P. Lévy. For systematic accounts, see, e.g., Bertoin [86], Sato [323], Rüdiger [320], and Applebaum [65]. For analytic aspects, see also Jacob [218–220] and Jacob and Schilling [222].

The nonstandard theory goes back to Lindstrøm [263] and Albeverio and Herzberg [14].

For further applications, see, e.g., Albeverio [2], Albeverio, Rüdiger, and Wu [54, 55] and Garroni and Menaldi [190].

Chapter 6

This chapter has a foundational purpose and is based on the monograph by Fajardo and Keisler [156], where also original work, in particular due to Hoover and Keisler [210], is discussed.

Appendix

Nonstandard analysis originated with the seminal work of Robinson [310,311] (with an early precursor by Schmieden and Laugwitz [325]). Further systematic expositions of nonstandard analysis include Machover and Hirschfeld [276], Bell and Machover [78], Albeverio, Fenstad, Høegh-Krohn, and Lind- strøm [25], Cutland [125], Davis [135], Stroyan and Bayod [341], Stroyan and Luxemburg [342] (for separate articles, see e.g., Albeverio, Luxem- burg, and Wolff [37], Arkeryd, Cutland, and Henson [68]). For more recent textbooks and monographs on nonstandard analysis, see e.g. Capiński and Cutland [102], Landers and Rogge [255], Goldblatt [194], Cutland [126], Loeb and Wolff [362], Ng [287], Sousa Pinto [335], and again Ng [289]. Fundamen- tal contributions to nonstandard measure-theoretic probability theory and nonstandard stochastic analysis were made, among others, by Loeb [267]and Anderson [61,62]. Other, hitherto unmentioned, examples of work on stochastic nonstandard analysis can be found in Albeverio and Fan [8], Albeverio, Luxemburg, and Wolff [37], Anderson [62], Arkeryd, Cutland, and Henson [68], Capinski and Cutland [102], to mention but a few; see, however, also the references in Albeverio et al. [25].

For expositions of Edward Nelson’s approach to nonstandard analysis (Internal Set Theory and an elementary subsystem thereof, called Radically Elementary Mathematics), we refer to Nelson [284–286], the introductory Historical Notes, Bibliographical Complements 261 contributions in Diener and Diener [144], and Robert [309]. Yet another approach to nonstandard analysis which promises to be of signiﬁcant ped- agogical value has been suggested by Benci and Di Nasso [83]. A comparison of diﬀerent approaches to nonstandard analysis can be found in Benci, Forti and Di Nasso [84].

Foundational questions of nonstandard analysis have been systematically studied in a monumental treatise by Kanovei and Reeken [229]. Early contributions to this ﬁeld of research include Hrbáček [211]; more recent advances are due to, e.g., Kanovei and Shelah [230].

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Notation Index

Logical notation:

∀–for all ∃–exists ∈–in ⊂–subset or equal ∅–empty set

Other notation: a.e.–almost every A–infinitesimal generator of E(·, ·) Aˆ–infinitesimal co-generator of E(·, ·) A–infinitesimal generator of E(·, ·) Bρ(x)–the open ball of radius ρ centered at x (δ) Capα (·)–α-capacity of hyperfinite Dirichlet form B–complement of a subset B ⊂ Rd or B ⊂ ∗Rd d ∗ν∈M ν–convolution of all measures ν on R contained in the set M (also used to denote convolutions of internal finitely-additive measures) D(E)–domain of E(·, ·) D(E)–domain of E(·, ·) D(E)–domain of E(·, ·) D(E)–domain of E(·, ·) D(F )–domain of F (·, ·) e(A)–energy of internal additive functional A e(A, B)–mutual energy of internal additive functionals A and B E(·, ·)–standard or quasi-regular Dirichlet forms E(·, ·)–hyperfinite quadratic form or hyperfinite Dirichlet form Eˆ(·, ·)–co-form of E(·, ·) E(·, ·)–symmetric part of E(·, ·) E˚(·, ·)–anti-symmetric part of E(·, ·) F (·, ·)–standard coercive closed form or Dirichlet forms

281 282 Notation Index

Gα–resolvent of hyperfinite quadratic forms ˆ Gα–co-resolvent of hyperfinite quadratic forms Γα(·)–α-capacity of standard Dirichlet form H–hyperfinite dimensional linear space with inner product ·, · generating norm || · || N = {1, 2, ···}–natural number set without zero N0 = {0, 1, 2, ···}–natural number set {Qt}–semigroup of hyperfinite Markov chain {Qˆt}–co-semigroup of hyperfinite Markov chain t {Q }–semigroup of E(·, ·) R–1-dimensional Euclidean space R+ =[0, ∞)–all nonnegative real numbers R>0 =(0, ∞)–all positive real numbers {Rα}–resolvent of standard Dirichlet forms ˆ {Rα}–co-resolvent of standard Dirichlet forms S = {s0,s1, ··· ,sN }–state space S0 = {s1,s2, ··· ,sN }–state space without trap s0 ∗ T = {kΔt|k ∈ N0}–nonstandard time-line X and Xˆ–dual hyperfinite Markov chains Index

A D

Adapted automorphism, 246 δ-co-exceptional, 72 Adapted equivalence, 244 δ-exceptional, 68 Adapted function, 244 δ0-exceptional irregularities, 138 Adapted-homogeneous space, 245 δ-finite energy integral, 92 Adapted probability space, 244 δ-hyperfinite energy integral, 92 Adapted-saturated space, 245 δ-internal additive functional, 103 Adapted space, 244 δ-internal martingale additive Adapted-universal space, 245 functional of finite energy, 111 Adjoint operator, 3 δ-internal multiplicative ℵ1-Saturation Principle, 251 functional, 120 α-capacity, 79, 80, 178 δ-nearstandardly concentrated, 151 Anti-symmetric part, 5 δ-quasi-continuous core, 155 δ-quasi-continuous semigroup, 158 C δ-quasi-everywhere, 111 δ-zero capacity, 86 Canonical subprocess, 123 Dirichlet form, 167 1-capacity, 168 Domain of E(·, ·), 7 Cauchy sequence, 17 Dual hyperfinite Markov chain, 38 Choquet S0-capacity, 85 Closed, 13, 17, 32, 35, 53 E Co-equilibrium α-potential, 77 Coercive closed form, 53 E1-Cauchy sequence, 35 Co-exceptional, 72 E1-Cauchy sequence, 17 Continuity constant, 29, 53 E-closed, 32 Co-resolvent of E(·, ·), 27 E-closed, 13 Co-resolvent of F (·, ·), 55 Energy, 104 Co-semigroup, 4 Energy measure, 105 Co-semigroup of (F (·, ·),D(F )), 55 Equilibrium α-potential, 77, 79

283 284 Index

Exceptional, 68 Hyperfinite weak sector condition, Exceptional irregularities, 138 29 Extension, 249 Hyperreals, 249 External, 251 I F Increments of a hyperfinite Lévy Finer, 170 process, 203 Finite, 251, 253 Infinitesimal, 251, 253 Finite energy, 79 Infinitesimal co-generator, 4, 40 Finite B(Y ) partition of Y , 170 Infinitesimal generator, 4, 40 First co-resolvent equation, 28 Inner standard part, 131, 160, 192 First resolvent equation, 28 Internal, 251 Internal algebra, 252 G Internal, finitely additive measure, 252 Generating triplet (of a Lévy Internal martingale additive process), 201 functionals, 111 Internal multiplicative functional, H 120

Hausdorff space, 36 L Hull of (H, ·, ·), 7 2 Hyperfinite adapted probability λ -martingale, 119, 214 space, 246 Lévy-Khintchine formula, 200, 202 Hyperfinite α-co-excessive, 74 Lévy measure, 201 Hyperfinite α-co-potential, 93 Lévy process, 200 Hyperfinite α-excessive, 74 Lifetime, 137, 138 Hyperfinite α-potential, 93 Loeb algebra, 252 Hyperfinite δ-quasi-continuous, 155 Loeb measure, 252 Hyperfinite δ-quasi-continuous core, 155 M Hyperfinite Dirichlet form, 49 Hyperfinite Lévy process, 203 Markov operator, 44 Hyperfinite Markov chain, 37, 38, Markov property, 43 131 Martingale, 111 Hyperfinite pre-α-co-excessive, 93 Modified standard part, 139, 192 Hyperfinite pre-α-excessive, 93 Monad, 68, 146, 249, 251 Hyperfinite quadratic co-form, 41 Mutual energy of e(A, B), 104 Hyperfinite quadratic form, 40 Hyperfinite quasi-continuous, 155 N Hyperfinite quasi-continuous core, 155 N-dimensional, 253 Hyperfinite random walk, 203 Nearstandard, 68, 253 Hyperfinite time-line, 246 Nearstandard point, 250 Hyperfinite weak coercive quadratic Nearstandardly concentrated, 151 form, 29 Nest, 168 Index 285

Non-anticipating, 103 S-dense, 53, 250 Nonnegative quadratic co-form, 4 Semi-compact, 85 Nonnegative quadratic form, 3 Semigroup, 4, 55, 120 Nonnegative symmetric quadratic Separates points, 146, 180 form, 6 Separating family, 146 Nonstandard reals, 249 Set of irregularities, 138 Nonstandard version or extension, S-left limit, 137 249 S-right limit, 137 ν-dual, 167 Splitting infinitesimal, 224 ν-tight, 166 S-rich, 250 Standard part, 68, 217, 220, 251 P Standard part of a hyperfinite Lévy process, 220 Paved set, 84 Standard part of E(·, ·), 34 Paving, 84 Standard part of point, 250 Pre-nearstandard, 253 Standard part of a set, 132 Principle of Countable Saturation or Standard part operation, 170 ℵ 1-Saturation, 251 Stopping time, 136 Properly associated, 168 Strong Markov process, 137 Properly co-exceptional, 72 Sub-Markovian operator, 167 Properly δ-co-exceptional, 72 Subordinate, 121 Properly δ-exceptional, 69 Subprocess, 122 Properly exceptional, 69 Superstructure, 252 Q Symmetric hyperfinite quadratic form, 41 Symmetric part E(·, ·), 5 Quasi-continuous, 168 Symmetric part of E(·, ·), 17 Quasi-continuous semigroup, 158 Symmetric standard part of Quasi-regular, 169 E(·, ·), 17 R T

Radon measure, 170 Truncated random walk, 206, 216 Regular Hausdorﬀ space, 145 T3-space, 145 Resolvent of E(·, ·), 27 Resolvent of E(·, ·), 18 U Resolvent of F (·, ·), 55 Rich, 170 Unit contraction, 43 Right continuous ﬁltration, 136 Universally measurable, 136 Right standard part, 138, 217, 220 W S Weak sector condition, 35, 53 S0-analytic, 85 Saturation Principle, 251 Z S-bounded, 7, 113 S-continuous, 113 Zero capacity, 86, 168 Editor in Chief: Franco Brezzi

Editorial Policy

1. The UMI Lecture Notes aim to report new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are also welcome.

2. Manuscripts should be submitted (preferably in duplicate) to Redazione Lecture Notes U.M.I. Dipartimento di Matematica Piazza Porta S. Donato 5 I – 40126 Bologna and possibly to one of the editors of the Board informing, in this case, the Redazione about the submission. In general, manuscripts will be sent out to external referees for evaluation. If a decision cannot yet be reached on the basis of the first 2 reports, further referees may be contacted. The author will be informed of this. A final decision to publish can be made only on the basis of the complete manuscript, however a refereeing process leading to a preliminary decision can be based on a pre-final or incomplete manuscript. The strict minimum amount of material that will be considered should include a detailed outline describing the planned contents of each chapter, a bibliography and several sample chapters.

3. Manuscripts should in general be submitted in English. Final manuscripts should contain at least 100 pages of mathematical text and should always include

– a table of contents; – an informative introduction, with adequate motivation and perhaps some historical remarks: it should be accessible to a reader not intimately familiar with the topic treated; – a subject index: as a rule this is genuinely helpful for the reader.

4. For evaluation purposes, manuscripts may be submitted in print or electronic form (print form is still preferred by most referees), in the latter case preferably as pdf- or zipped ps-ﬁles. Authors are asked, if their manuscript is accepted for publication, to use the LaTeX2e style ﬁles available from Springer’s web-server at ftp://ftp.springer.de/pub/tex/latex/svmonot1/ for monographs and at ftp://ftp.springer.de/pub/tex/latex/svmultt1/ for multi-authored volumes

5. Authors receive a total of 50 free copies of their volume, but no royalties. They are entitled to a discount of 33.3% on the price of Springer books purchased for their personal use, if ordering directly from Springer.

6. Commitment to publish is made by letter of intent rather than by signing a formal contract. Springer-Verlag secures the copyright for each volume. Authors are free to reuse material contained in their LNM volumes in later publications: A brief written (or e-mail) request for formal permission is sufﬁcient.