A Class of Infinite Dimensional Stochastic

Linköping Studies in Science and Technology, Licentiate Thesis No. 1612

A class of inﬁnite dimensional stochastic processes with unbounded diffusion

John Karlsson

Department of Mathematics Linköping University, SE–581 83 Linköping, Sweden Linköping 2013 Linköping Studies in Science and Technology, Licentiate Thesis No. 1612

A class of inﬁnite dimensional stochastic processes with unbounded diffusion John Karlsson

[email protected] www.mai.liu.se Mathematical Statistics Department of Mathematics Linköping University SE–581 83 Linköping Sweden

LIU-TEK-LIC-2013:46 ISBN 978-91-7519-536-0 ISSN 0280-7971

Printed by LiU-Tryck, Linköping, Sweden 2013, 2nd printing

This is a Swedish Licentiate Thesis. The Licentiate degree comprises 120 ECTS credits of postgraduate studies.

Abstract

The aim of this work is to provide an introduction into the theory of infinite dimensional stochastic processes. The thesis contains the paper A class of infinite dimensional stochastic processes with unbounded diffusion written at Linköping University during 2012. The aim of that paper is to take results from the finite dimensional theory into the infinite dimensional case. This is done via the means of a coordinate representation. It is shown that for a certain kind of Dirichlet form with unbounded diffusion, we have properties such as closability, quasi-regularity, and existence of local first and second moment of the associated process. The starting chapters of this thesis contain the prerequisite theory for understanding the paper. It is my hope that any reader unfamiliar with the subject will find this thesis useful, as an introduction to the field of infinite dimensional processes.

Populärvetenskaplig sammanfattning

En stokastisk process är något som beskriver hur ett slumpmässigt system ändrar sig över tiden, t.ex. hur antalet kunder i en affär ändras under dagen eller hur värdet av en aktie för- ändras. Stokastiska processer har studerats i över 100 år och det ﬁnns många olika typer av dem. En av de enklaste typerna är Markovprocessen som är en slumpmässig process med ”dåligt minne”. Nästa värde som processen har beror bara på dess nuvarande tillstånd. Markovprocesser har studerats sedan början av 1900-talet när A. A. Markov studerade hur konsonanter och vokaler följer efter varandra i texter. En Markovprocess i kontinuerlig tid (d.v.s. inga hopp) som följer en kontinuerlig kurva kallas för en diffusionsprocess. En process kan ta värden i ﬂera dimensioner, exempelvis kan en partikels bana i rummet sägas vara tredimensionell. Den här avhandlingen behandlar oändligdimensionella processer d.v.s. processer som tar värden i rum med oändlig dimension. Man har visat att en viss typ av matematiskt objekt, en så kallad Dirichletform, ibland beskriver en sådan process. Den här avhandlingen studerar en speciell sorts Dirichletform och hittar krav för att vi ska kunna associera den med en process. Det är möjligt att dessa oändligdimensionella processer har viktig tillämpning inom teoretisk fysik.

vii

Contents

1 Introduction 1 1.1 Background ...... 1 1.2 Outline ...... 2

2 Chapter 2-Malliavin Calculus 5 2.1 General framework ...... 5 2.2 Hermite polynomials ...... 6 2.3 Wiener Chaos ...... 6 2.4 Multiple stochastic integrals ...... 8 2.5 Derivative operator ...... 11 2.6 Divergence operator ...... 15 2.7 The generator of the Ornstein-Uhlenbeck semigroup ...... 19

3 Chapter 3-Dirichlet form theory 21 3.1 Introduction ...... 21 3.2 Resolvents ...... 21 3.3 Semi-groups ...... 22 3.4 Bilinear forms ...... 23 3.5 Dirichlet forms ...... 24

4 Chapter 4-Main results 29 4.1 Earlier work ...... 29 4.2 Introduction ...... 31 4.3 Formal deﬁnitions ...... 32 4.4 Deﬁnition of the form ...... 33 4.5 Closability ...... 33 4.6 Quasi-regularity ...... 39

ix x Contents

4.7 The generator ...... 41 4.8 Local ﬁrst moment ...... 43

5 Chapter 5-Notation and basic deﬁnitions 47 5.1 Introduction ...... 47 5.2 Deﬁnitions ...... 47 5.3 Notation ...... 48

Bibliography 51 1 Introduction

HE aim of this work is to provide an introduction into the theory of infinite dimen- T sional stochastic processes. The thesis contains the paper A class of infinite dimensional stochastic processes with unbounded diffusion, a joint work with Jörg-Uwe Löbus, written at Linköping University during 2012. The starting chapters of the thesis contain the prerequisite theory for understanding the paper and it is my hope that any reader unfamiliar with the subject will find this useful as an introduction to the field.

1.1 Background

The theory of stochastic processes goes back to the beginning of the 20th century, and in the middle of the 20th century many important concepts were formed. An excellent overview of this is the article of P.-A. Meyer [6]. Doob worked on Martingales in the time period 1940-1950 and in 1945 a paper was published detailing the strong Markov property. Itô had his first article about the stochastic integral published in 1944 which clearly has had a large impact on current theory. In 1948 Hille and Yoshida formulated the theorem describing the semigroups of continuous operators. This had a large impact on Markov theory since the nature of a Markov process provides a natural connection to semigroups. Markov processes are of interest due to their natural occurence. As men- tioned, because the evolution of a Markov process only depends on its present state, any observation in nature could be considered Markov provided there is sufficient informa- tion. If the time is discrete we call it a Markov chain. While Markov chains change value at discrete times a diffusion is a Markov process with continuous trajectories. From Kolmogorov we know that for a Markov process, the transition function is usually a solution of the Focker-Planck parabolic partial differential equation. In 1959 Beurling and Deny introduced the theory of Dirichlet forms. Their paper was in the area of potential theory. This theory was brought into the realm of probablity by Silverstein in 1974 and Fukushima in 1975. This is a useful tool for creating Markov processes in infinite dimen-

1 2 1 Introduction sional spaces, something that might have applications in physics. In 1976 P. Malliavin introduced the theory of Malliavin calculus, first developed to prove a theorem in the theory of partial differential equations. From the Malliavin calculus comes the Wiener chaos and Ornstein-Uhlenbeck operator which can be used for analysis on the Wiener space and thus allowing analysis on infinite dimensions, once again with possible applications in theoretical physics. In 1992 B. Driver and M. Röckner [2] published a paper detailing the construction of an "Ornstein-Uhlenbeck" like process on the path space of a compact Riemannian manifold without boundary. In 2004 J.-U. Löbus published a paper [4] describing a class of processes on the path space of a Riemannian manifold. In this paper the operator describing the diffusion was unbounded in contrast to earlier papers. F.-Y. Wang and B. Wu generalized the results in these papers and showed that the Riemannian manifold can be noncompact, in addition they extended the class of Dirichlet forms studied. Their paper [11] was published in 2008. The same authors published another paper [12] in 2009 detailing a large class of quasi-regular Dirichlet forms with unbounded, and non-constant diffusion coefficients on free Riemannian path spaces.

1.2 Outline

This thesis consists of ﬁve chapters and the outline is as follows.

Chapter 1 This chapter contains a historical overview of the theory. It also contains the purpose of this work as well as providing an outline of the layout of the thesis.

Chapter 2 Chapter two presents the basics of Malliavin Calculus. The chapter begins with introducing the Hermite polynomials as a starting point for the Wiener chaos decomposition. It is then shown that this decomposition has a representation in the form of Itô integrals. We then present the deﬁnition of the Malliavin gradient D and some of its properties, also connecting it to the Wiener chaos decomposition. The end of the chapter contains a presentation of the divergence operator δ, the generator of the Ornstein-Uhlenbeck semigroup L, and the relationship between D, δ and L.

Chapter 3 Chapter three is an introduction to the theory of Dirichlet forms. The first part of the chapter recalls the definitions of resolvents, semigroups, and the infinitesimal generator associated to them. We also state the famous Hille–Yosida theorem for contraction semigroups and by that note the connection between semigroup, resolvent, and generator. The second part of the chapter connects the first part to bilinear forms. The last part of the chapter states the definition of a Dirichlet form and quasi-continuity. We also state the 1.2 Outline 3 theorem connecting the Dirichlet form to a stochastic process. The section ends with stating the famous Cameron–Martin theorem.

Chapter 4 Chapter four contains the paper A class of inﬁnite dimensional stochastic processes with unbounded diffusion written at Linköping University during 2012. The version here is more detailed than the preprint version at ArXiv. The paper is a study of a particular kind of Dirichlet form. We study the form on cylindrical functions and under some assumptions we prove the closability of the form. The form is shown to be quasi-regular. The last parts of the paper studies the local ﬁrst and second moment of the associated process.

Chapter 5 Chapter ﬁve contains elementary deﬁnitions of concepts used in the thesis. The section also contains a list of notation and abbreviations used throughout this work.

2 Chapter 2-Malliavin Calculus

HE Malliavin calculus, named after the French matematician Paul Malliavin, extends T the calculus of variations, from functions to stochastic processes. In particular it makes it possible to calculate the derivative of random variables. This section serves to introduce the notions used in the ﬁeld of Malliavin calculus. The section is heavily based on David Nualart’s monographs [8] and [7].

2.1 General framework

Let H be a real separable Hilbert space with inner product h·, ·iH and corresponding norm 2 k · kH. In this presentation the Hilbert space H will mostly be considered to be L ([0, 1]) with the Borel σ-algebra B and Lebesgue measure µ, while in a more general setting it would be L2(T, B, µ) where (T, B) is a measurable space and µ is an atomless measure on (T, B).

Deﬁnition 2.1.1. A stochastic process W = {W (h), h ∈ H} deﬁned in a complete probability space (Ω, F,P ) is a Gaussian process on H, if W is a centered Gaussian family of random variables and E[W (g)W (h)] = hg, hiH for all g, h ∈ H. This is illustrated by a simple example. Example 2.1 2 Let H = L (R+) and Wt := W (1[0, t]), t ≥ 0. This is standard Brownian motion since

E[WsWt] = E[W (1[0,s])W (1[0,t])] = h1[0,s], 1[0,t]iL2 = min(s, t).

Using the same notation as in Nualart [8] letting H1 := W , we note that H1 is a closed

5 6 2 Chapter 2-Malliavin Calculus

2 subspace of L (Ω, F,P ), and that W : H → H1 deﬁnes a linear isometry between H and H1. We will denote by W (A) := W (1A). Given a Hilbert space H, a probability space and Gaussian process satisfying the above conditions exist. This is a consequence of Kolmogorov’s extension theorem.

2.2 Hermite polynomials

The Hermite polynomials will play an important role in the upcoming Wiener chaos decomposition of a Gaussian random variable. The Hermite polynomials Hn are the coefﬁ- cients of the Taylor expansion of F (x, t) = exp{tx − t2/2} in powers of t, that is: ∞ 2 tx− t X n F (x, t) := e 2 = Hn(x) · t . (2.1) n=0 We have the relations:

n 2 n 2 1 2 (−1) x d − x H (x) = 1,H (x) = x, H (x) = (x − 1),H (x) = e 2 e 2 0 1 2 2 n n! dxn and also: d H (x) = H (x), (n + 1)H (x) = xH (x) − H (x) dx n n−1 n+1 n n−1 as well as: ( 0 if n odd, Hn(0) = (−1)k 2kk! if n even. Remark 2.2.1. The term Hermite polynomial does not have a strict deﬁnition. In other literature the polynomials H˜ or Hˆ deﬁned by ∞ 2 n tx− t X t e 2 = H˜ (x) · n n! n=0 or ∞ 2 n 2tx− t X t e 2 = Hˆ (x) · n n! n=0 are classifed as the Hermite polynomials.

2.3 Wiener Chaos

2 Assume that H (we may think of it as L ([0, 1])) is infinite dimensional and let {e1, e2,... } be an ON-basis in H. We let Λ denote the set of all finite multi-indices a = (a1, a2,... ), ai ∈ N for i = 1, 2,... , i.e. only finitely many ai:s are non-zero. For any a ∈ Λ we define: ∞ ∞ ∞ Y X √ Y a! := ai!, |a| := ai, Φa := a! Hai (W (ei)). i=1 i=1 i=1 Gaussian variables and Hermite polynomials are connected. The next lemma is related to orthogonality in L2(Ω, F,P ). 2.3 Wiener Chaos 7

Lemma 2.3.1. If (X,Y ) is a two-dimensional Gaussian vector s.t. E[X] = E[Y ] = 0 and V [X] = V [Y ] = 1. Then for all m, n ∈ N we have: ( 0 if m 6= n, E[Hm(X)Hn(Y )] = 1 n n! E[XY ] if m = n.

Z 1 E[Z2] Proof: For a Gaussian random variable Z we have E[e ] = e 2 and thus

sX+tY 1 E[(sX+tY )2] 1 E[s2X2] 1 E[t2Y 2] E[stXY ] E[e ] = e 2 = e 2 e 2 e 1 s2 1 t2 E[stXY ] = e 2 e 2 e , and we get ∞ ∞ 2 2 stE[XY ] sX− s tY − t X m X n e = E[e 2 e 2 ] = E[( s (Hm(X))( t Hn(Y ))]. m=0 n=0 Taylor expansion of the left hand side gives: s2t2E[XY ]2 sntnE[XY ]2 estE[XY ] = 1 + stE[XY ] + + ··· + + ... 2! n! and identifying coefﬁcients of smtn yields: ( 0 if m 6= n, E[Hm(X)Hn(Y )] = 1 n n! E[XY ] if m = n.

Using this lemma we can proceed to prove that the Hermite polynomials can be used to construct an ON-basis for L2(Ω, F,P ). 2 Proposition 2.3.2. The collection {Φa : a ∈ Λ} is an ON-basis for L (Ω, F,P ).

Proof: Step 1: We begin by showing that {Φa : a ∈ Λ} forms an orthonormal system. Let a, b ∈ Λ. Since W (ei) and W (ej) are independent for i 6= j we get: ∞ ∞ √ Y √ Y E[ΦaΦb] = E[ a! Hai (W (ei)) · a! Hbi (W (ei))] i=1 i=1 ∞ ( Y 1 if a = b, = a! · E[Ha (W (ei))Hb (W (ei))] = i i 0 . i=1 otherwise

Step 2: We show that {Φa : a ∈ Λ} forms a complete system. Assume E[XΦn] = 0 for all a ∈ Λ for some X ∈ L2(Ω, F P ) where X 6= 0. This means that X is orthogonal to every polynomial in the variables W (e1),...,W (en) for all n ∈ N. We get Pn i λj W (ej ) E[Xe j=1 ] = 0, ∀ λ1, . . . , λn ∈ R, n ∈ N. (2.2)

The left hand side is the Fourier transform of E[X|W (e1),...,W (en)] and consequently E[X|W (e1),...,W (en)] = 0 for all n ∈ N. Since F is generated by W (h), h ∈ H and e1, e2,... is an ON-basis in H it follows that X = 0, thus {Φa : a ∈ Λ} must be a complete system. 8 2 Chapter 2-Malliavin Calculus

2 We let Hn denote the closed subspace of L (Ω, F,P ) spanned by {Φa : a ∈ Λ, |a| = 2 L∞ n}. It follows that L (Ω, F,P ) = n=0 Hn. Hn is called the nth Wiener Chaos. We denote by Jn the projection operator onto the nth Wiener Chaos, i.e.

∞ X F = JnF. n=0 Remark 2.3.3. The nth Wiener chaos contains polynomials of Gaussian variables of degree n.

∞ We arrived at the nth Wiener chaos as an expression of our ON-basis (ei)i=1. How- ever the following proposition shows that the nth Wiener chaos is independent of the choice of this ON-basis.

Proposition 2.3.4. For each n ≥ 0 the set of random variables

{Hn(W (h)) : h ∈ H, khkH = 1} is dense in Hn.

2 Proof: The linear span of {Hn(W (h)) : h ∈ H, n ≥ 0} is dense in L (Ω, F,P ) since if a random variable X is orthogonal to this set, it is orthogonal to the span of the sets {W (h)n : h ∈ H, n ≥ 0}. Following the procedure of Proposition 2.3.2 we see that in 0 this case X = 0. Letting Hn denote the closed span of the family {Hn(W (h)) :, h ∈ H} 2 L∞ 0 LN 0 we get L (Ω, F,P ) = n=0 Hn. We have that n=0 Hn coincides with the set of polynomials of the family {W (h): h ∈ H} with degree less than or equal to N. We LN also have that n=0 Hn is the span of the set of polynomials {W (e1),W (e2),... } of ∞ L∞ degree less than or equal to N. Since (ei)i=1 is an ON-basis it follows that n=0 Hn = L∞ 0 n=0 Hn and the claim follows.

2.4 Multiple stochastic integrals

We will see that there is another representation for the nth Wiener chaos in the form of iterated Itô integrals. We begin with introducing a type of elementary functions ﬁrst described by Norbert Wiener.

Deﬁnition 2.4.1. We deﬁne En as the set of all elementary functions of form

k X f(t , . . . , t ) = a 1 (t , . . . , t ), (2.3) 1 n i1...in Ai1 ×···×Ain 1 n i1,...,in=1 where ai1...in ∈ R and ai1...in = 0 if two indices coincide and A1,...,Ak are pairwise disjoint subsets of [0, 1] with ﬁnite Lebesgue measure.

We see that this is the usual simple functions in [0, 1]n with the condition of being 0 on any diagonal described by two equal coordinates. It can be shown that this set has measure 0 and we get the following result. 2.4 Multiple stochastic integrals 9

2 n Proposition 2.4.2. The set of elementary functions En is dense in L ([0, 1] ).

2 n Proof: We show that any characteristic function 1A in L ([0, 1] ) where A = A1 ×· · ·× An, A1,...,An ∈ B([0, 1]) can be approximated by a function in En. Given ε > 0 we can ﬁnd B1,...,Bk ∈ B([0, 1]) such that |Bi| < ε, i = 1, . . . , k, the sets Bi are disjoint, and every Ai can be written as a union of a collection of the sets Bi. We have

k X 1 = a 1 A i1,...,in Bi1 ×···×Bin i1,...,in=1 where ai1,...,in is either 0 or 1 for each multiindex i. We divide this sum into two parts. We let I be the set of n-tuples (i1, . . . , in) where all the indices are different and J be the remaining set of n-tuples. We deﬁne

k X 1 = a 1 . B i1,...,in Bi1 ×···×Bin (i1,...,in)∈I

Now IB belongs to En and we have X 1 1 2 n 1 k A − BkL ([0,1] ) = ai1...in Bi ×···×Bi 1 n L2([0,1]n) (i1,...,in)∈J X = ai1...in µ(Bi1 ) · ... · µ(Bin )

(i1,...,in)∈J k k n X X k−2 ≤ µ(B )2 µ(B ) 2 i i i=1 i=1 n where the last inequality follows from the facts that 2 is loosely the number of diago- Pk k−2 nals, µ(Bi) is the thickness of it and i=1 µ(Bi) is an upper bound of its length. Since Bi is in B[0, 1], i = 1, . . . , k and µ(Bi) < ε we get

k k k n X X k−2 n X k−1 n µ(B )2 µ(B ) ≤ ε µ(B ) ≤ ε. 2 i i 2 i 2 i=1 i=1 i=1 Now since we can approximate characteristic functions it follows that simple functions can be approximated. As simple functions are dense in L2 the statement follows. We can now deﬁne a stochastic integral for any elementary function. For a function of the form (2.3) we deﬁne

k X In(f(t1, . . . , tn)) = ai1,...,in W (Ai1 ) · ... · W (Ain ).

i1,...,in=1

Remark 2.4.3. In(f) can be expressed as an iterated Itô integral

1 t t Z Z n Z 2

In(f) = n! ... f(t1, . . . , tn) dWt1 . . . dWtn .

tn=0 tn−1=0 t1=0 10 2 Chapter 2-Malliavin Calculus

The following proposition shows some of the properties of this stochastic integral.

Proposition 2.4.4. In has the following properties: ˜ ˜ (i) In(f) = In(f), where f denotes the symmetrization of f, i.e.

1 X f˜(t , . . . , t ) = f(t , . . . , t ) 1 n n! Π(1) Π(n) Π where Π runs over all permutations of {1, . . . , n}. (ii) ( 0 if m 6= n E[Im(f),In(g)] = ˜ n!hf, g˜iL2([0,1]n) if m = n.

Proof: (i) Assume that f has the form in (2.3), i.e.

k X f(t , . . . , t ) = a 1 (t , . . . , t ). 1 n i1...in Ai1 ×···×Ain 1 n i1,...,in=1

The property follows immediately since

k X f(t , . . . , t ) = a 1 (t , . . . , t ) Π(1) Π(n) Π(i1)...Π(in) AΠ(i1)×···×AΠ(in) 1 n i1,...,in=1 k X = a 1 (t , . . . , t ). i1...in Ai1 ×···×Ain 1 n i1,...,in=1

(ii) We may assume that f and g are both associated with the same partition A1,...,Ak otherwise we may reﬁne the partition to satisfy the assumption. Let

k X g(t , . . . , t ) := b 1 (t , . . . , t ). 1 n i1...in Ai1 ×···×Ain 1 n i1,...,in=1

Assume m 6= n. Using (i) we get

˜ E[Im(f)In(g)] = E[Im(f)In(˜g)] " k # X ˜ = E a˜i1...im bj1...jn W (Ai1 ) · ... · W (Ajn ) = 0

i1,...,im=1, j1,...,jn=1 2.5 Derivative operator 11

since at least two of the sets Ai1 ...Ajn are disjoint. Now for m = n we get ˜ E[In(f)In(g)] = E[Im(f)In(˜g)] " # X 2 ˜ = E (n!) a˜i1...in bi1...in W (Ai1 ) · ... · W (Ain )

i1,...,,in=1 ˜ = n!hf, g˜iL2([0,1]n).

2 n ˜ 2 n It follows that we get an isometry between Ls([0, 1] ) := {f : f ∈ L ([0, 1] )} and Hn, i.e. √ ˜ n!kfkL2([0,1]n) = kI(f)kL2(Ω,F,P ). 2 n Since En is dense in L ([0, 1] ) we can extend In to a linear continuous operator from L2([0, 1]n) to L2(Ω, F,P ).

2 n Remark 2.4.5. The image of Ls([0, 1] ) under In is the nth Wiener chaos Hn since In(f) is a polynomial of degree n in W (A1),...,W (An) when f has the form (2.3), and Proposition 2.4.4(ii) shows that stochastic integrals of different order are orthogonal. The claim then follows from induction.

Since we can express a random variable in its chaos expansion we note the following:

2 2 n Remark 2.4.6. If F ∈ L (Ω, F,P ), then there exists fn ∈ Ls([0, 1] ), n ∈ N such that

∞ X F = E[F ] + In(fn). n=1

2.5 Derivative operator

To deﬁne the calculus on the Wiener space a natural way of proceeding is to ﬁrst look at a simple class of functions. Let Z denote the set of smooth cylindrical functions

∞ Z = {F : F = f(W (h1),...,W (hn)), h1, . . . , hn ∈ H, f ∈ Cp }

∞ where Cp denotes smooth functions with polynomial growth. Deﬁnition 2.5.1. The gradient of a smooth cylindrical function is deﬁned by:

n X ∂f DF := (W (h ),...,W (h )) · h . ∂x 1 n i n=1 i 12 2 Chapter 2-Malliavin Calculus

Remark 2.5.2. We have that DW (h) = h. Clearly we have the product rule D(FG) = DF · G + F · DG and we also get an integration by parts formula as will be shown shortly. We also deﬁne the directional derivative in the following manner.

Deﬁnition 2.5.3. The directional derivative Dh is deﬁned by

DhF := hDF, hi h ∈ H. This deﬁnition of the gradient D gives the following integration by parts formula

Proposition 2.5.4. Suppose F ∈ Z and h ∈ H. Then E[hDF, hiH] = E[F · W (h)]. ∞ Proof: First we assume h = e1 and F = f(W (e1),...,W (en)) where f ∈ Cp . Now let Φ denote the density of a standard n-dimensional normal distribution, i.e.

( n ) 1 X Φ(x , . . . , x ) = (2π)−n/2 exp − x2 . 1 n 2 i i=1 We have ∂f Z ∂f E[hDF, hiH] = E[ (W (e1),...,W (en))] = Φ dx ∂x1 ∂x1 n R Z = fΦx1 dx = E[F · W (e1)]) = E[F · W (h)].

n R A change of basis proves it in the general case.

We would like to extend D to an operator from L2(Ω, F,P ) to L2(Ω, F,P ; H) and do so by ﬁrst stating the next lemma.

Lemma 2.5.5. Let F,G ∈ Z and h ∈ H. Then

E[GhDF, hiH] = E[−F hDG, hiH + F GW (h)]. Proof: Using the product rule and Proposition 2.5.4 we get:

E[GhDF, hiH] = E[hG · DF, hiH] = E[hD(FG) − F · DG, hiH]

= E[−F hDG, hiH] + E[hD(FG), hiH]

= E[−F hDG, hiH] + E[FG · W (h)].

Using this lemma we prove the following proposition.

Proposition 2.5.6. The operator D is closable from Lp(Ω, F,P ) to Lp(Ω, F,P ; H) for any p ≥ 1.

Proof: Let Fn ∈ Z, n = 1, 2,... such that: 2.5 Derivative operator 13

p • Fn → 0 in L ,

p • DFn is Cauchy in L (Ω, F,P ; H).

Let η := limn→∞ DFn. Left to prove is η = 0. Let h ∈ H and G = g(W (h1),...,W (hn)), ∞ n g ∈ Cb (R ) be any smooth random variable s.t. GW (h) is bounded. These G are dense in Lp(Ω, F,P ; H). It follows using the above lemma

E[hη, hi G] = lim E[hDFn, hi G] H n→∞ H

= lim E[−FnhDG, hi + FnGW (h)] = 0. n→∞ H

It follows hη, hiH = 0 and since h was arbitrary we get η = 0. As a consequence there exists a unique closed extension (D, D1,p) of (D,Z). Here the case p = 2 is mostly considered. The closability of D gives rise to the chain rule

Proposition 2.5.7. (Chain rule) Let ϕ : Rm → R be a continuously differentiable func- 1,p m tion with bounded partial derivatives. Suppose F = (F1,...,Fm) ∈ (D ) for some p ≥ 1. Then ϕ(F ) ∈ D1,p and m X ∂ϕ D(ϕ(F )) = (F )DF . ∂x i i=1 i

1,p (n) Proof: Since D is closed we can ﬁnd sequences Fi ∈ Z, i = 1, . . . , m such that (n) p (n) p F −→ Fi, in L , and DF −→ DFi, in L (Ω, F,P ; H). The claim follows. i n→∞ i n→∞ Taking the gradient of a function lowers the corresponding degree with respect to the Wiener chaos in a similar fashion as taking the derivative of a polynomial lowers the degree by one. The following two propositions describe this relation in detail. 2 P∞ Proposition 2.5.8. Let F ∈ L (Ω, F,P ) with F = i=0 JnF corresponding to the Wiener chaos decomposition. Then F ∈ D1,2 if and only if ∞ X E kDF k2 = nkJ F k2 < ∞. H n 2 n=1 In this case, for all n ∈ N and h ∈ H we have

hD(JnF ), hiH = Jn−1 (hDF, hiH) . Proposition 2.5.9. Let F ∈ D1,2 and assume that F has representation ∞ X F = E[F ] + In(fn) n=1

2 n where fn ∈ Ls([0, 1] ). Then ∞ X DtF = nIn−1(fn(·, t)), t ∈ [0, 1]. n=1 14 2 Chapter 2-Malliavin Calculus

Proof: First let F = In(fn) where

k X f (t , . . . , t ) = a 1 (t , . . . , t ). n 1 n i1...in Ai1 ×···×Ain 1 n i1,...,in

Then k X F = In(fn) = ai1...in W (Ai1 ) · ... · W (Ain )

i1,...,in=1 and F ∈ Z. We get

n k X X D F = a W (A ) · ... · 1 (t) · ... · W (A ) = nI (f(·, t)) t i1...in i1 Aij in n−1 j=1 i1,...,in=1 by symmetry. The claim follows from the closedness of D1,2.

Suppose that F is given. How does one ﬁnd the representation of F ? The following proposition provides a tool to achieve this.

Proposition 2.5.10. If F ∈ D∞,2 and

∞ X F = E[F ] + In(fn) n=1 then 1 f = E[DkF ],K ∈ . k k! N

Proof: Applying Proposition 2.5.9 k times yields

∞ X Dk F = n(n − 1) ... (n − k + 1)I (f (t , . . . , t ; ·)) t1,...,tk n−k n 1 k n=k and thus

∞ ∞ 1 k X X D F = I0(fk(t1, . . . , tk)) + In(gn) = fk(t1, . . . , tk) + In(gn) k! t1,...,tk n=1 n=1

g 1 Dk F where n is the representing function of the chaos expansion of k! t1,...,tk . It follows that 1 k E[D F ] = fk(t1, . . . , tk). k! t1,...,tk

The following example illustrates the usage of the above proposition. 2.6 Divergence operator 15

Example 2.2

Let Wt, t ∈ [0, 1] be a one dimensional Wiener process. The Wiener chaos expansion of 2 F = Wa W1, 0 ≤ a ≤ 1 is then

a t t Z Z 3 Z 2 2 Wa W1 = 2aWa + aW1 + 4 dWt1 dWt2 dWt3

t3=0 t2=0 t1=0 1 t t Z Z 3 Z 2 1 + 2 [0,a](t1 ∨ t2) dWt1 dWt2 dWt3 .

t3=0 t2=0 t1=0 2 1 2 1 We note that Wa W1 = W ( [0,a]) W ( [0,1]) and use Proposition 2.5.10 to obtain

2 f0 = E[Wa W1] = 0,

2 2 f1(t1) = E[Dt1 Wa W1] = E[DWa W1]t1 2 = E[2W (1[0,a])W (1[0,1])1[0,a](t1) + W (1[0,a]) 1[0,1](t1)] 1 21 1 = 2E[WaW1 [0,a](t1) + Wa [0,1](t1)] = 2a [0,a](t1) + a,

1 2 2 f2(t1, t2) = E[D W W1] = ... 2 t1,t2 a = E[W1]1[0,a](t1 ∨ t2) + E[Wa] 1[0,1](t1) + 1[0,a](t2) = 0, where the ... signiﬁes using the elementary operations as for calculation of f1. In the same manner we get 1 f (t , t , t ) = E[D (D (D W 2W ))] = ... 3 1 2 3 3! t1 t2 t3 a 1 1 = 1 (t ∨ t ) + 1 (t ∨ t ) + 1 (t ∨ t ) . 3 [0,a] 2 3 [0,a] 1 2 [0,a] 1 3 We recall

1 1 t t Z Z Z 3 Z 2

F = f1(t1) dWt1 + 3! f3(t1, t2, t3) dWt1 dWt2 dWt3

0 t3=0 t2=0 t1=0 and the statement follows.

2.6 Divergence operator

We let Dom δ be the set of all u ∈ L2(Ω, F,P ; H) such that there exists c(u) > 0 with

|E[hDF, uiH]| ≤ ckF kL2 (2.4) 16 2 Chapter 2-Malliavin Calculus

1,2 1,2 for all F ∈ D . Then E[hDF, uiH] is a bounded linear functional from D to R. Now since D1,2 is dense in L2(Ω, F,P ) Riesz representation theorem states that there exists a unique representing element δ(u), bounded in L2(Ω, F,P ). That is

1,2 E[hDF, uiH] = E[F δ(u)], ∀ F ∈ D . (2.5)

In other words δ : L2(Ω, F,P ; H) → L2 and δ is the adjoint operator to D. We call δ the divergence operator.

Proposition 2.6.1. The divergence operator (δ, Dom δ) has the following properties (i)

E[δ(u)] = 0 ∀ u ∈ Dom δ.

n Pn o (ii) Suppose u ∈ ZH := v = j=1 Fjhj : Fj ∈ Z, hj ∈ H , then u ∈ Dom δ and

n n X X δ(u) = FjW (hj) − hDFj, hjiH. j=1 j=1

(iii) The operator (δ, Dom δ) is closed, i.e. if un ∈ Dom δ, n = 1, 2,... , un → u in 2 2 L (Ω, F,P ; H), and δ(un) → G in L , then u ∈ Dom δ and δ(u) = G.

Proof: (i) This follows from putting F ≡ 1 in (2.5). (ii) Using Lemma 2.5.5 we have   n X |E[hDF, ui ]| = E  FjhDF, hji  H H j=1    n n X X = E F  −hDFj, hji + FjW (hj) ≤ c(u)kF kL2 H j=1 j=1 so u ∈ Dom δ. We also get

  n n  X X E[hDF, uiH] = E F  −hDFj, hjiH + FjW (hj) = E[F δ(u)] j=1 j=1 and the statement follows. 2 1,2 (iii) Let un → u in L (Ω, F,P )). Then for any F ∈ D we have

E[F δ(un)] = E[hDF, uni ] −→ E[hDF, ui ] = E[F δ(u)]. H n→∞ H

Since F is arbitrary it follows that G = δ(u), and also u ∈ Dom δ since (2.4) is satisﬁed with c(u) = kGkL2 .

The following formulas state the relation between divergence δ and gradient D. 2.6 Divergence operator 17

Proposition 2.6.2. Let u, v ∈ ZH, F ∈ Z and h ∈ H. Let {ei : i ∈ N} be an ON-basis in H. Then

a) Dh(δ(u)) = hu, hiH + δ(Dh(u)).  ∞  X b) E[δ(u)δ(v)] = E[hu, viH] + E  Dei hu, ejiHDej hv, eiiH . i,j=1

c) δ(F u) = F δ(u) − hDF, uiH. Pn Proof: a) Let u = j=1 Fjhj where FjZ, hj ∈ H, j = 1, . . . , n. Now

Proposition 2.6.1(ii) z n }| n { X X Dh(δ(u)) = hD(δ(u)), hiH = D FjW (hj) − hDFj, hjiH , h j=1 j=1 H * n n n + X X X = FjDW (hj) + DFjW (hj) − D hDFj, hjiH , h j=1 j=1 j=1 H n n n X X X = Fjhh, hjiH + DhFjW (hj) − Dh hDFj, hjiH j=1 j=1 j=1 n n X X = hu, hiH + (DhFj)W (hj) − hD(DhFj), hjiH) j=1 j=1

= hu, hiH + δ(Dh(u)) where we in the last line once again used Proposition 2.6.1(ii). b) Using a) we obtain

" ∞ # X E[δ(u)δ(v)] = E[hu, Dδ(v)iH] = E hu, eiiHDei δ(v) i=1 " ∞ # X = E hu, eiiH hv, eiiH + δ Dei (v) i=1 ∞ X = E [hu, viH] + E hu, eiiHδ hDv, eiiH) i=1  ∞ * ∞ ∞ !+  X X X = E [hu, viH] + E  Dej hu, eiiHej,Dei hv, ekiek  i=1 j=1 k=1 H  ∞  X = E [hu, viH] + E  Dej hu, eiiH · Dei hv, ejiH . i,j=1 18 2 Chapter 2-Malliavin Calculus

c) For G ∈ Z we have

E[Gδ(F u)] = E[hDG, F uiH = E[hF DG, uiH] = E[hu, D(FG) − GDF iH]

= E[(δ(u)F − hu, DF iH)G].

Since G was arbitrary it follows δ(F u) = F δ(u) − hDF, uiH. We state the connection between the Wiener chaos expansion and the divergence operator. Recall that for u ∈ L2([0, 1] × Ω), u has a Wiener chaos expansion of form

∞ X u(t) = In(fn(·, t)), n=0

2 n where fn ∈ L ([0, 1] ) and fn is symmetric in the n ﬁrst variables. Proposition 2.6.3. We have u ∈ Dom δ if and only if the series

∞ X ˜ In+1(fn) n=0 converges in L2(Ω, F,P ). In this case we have

∞ X ˜ δ(u) = In+1(fn). n=0

2 n Proof: Suppose G = In(g) where g ∈ Ls([0, 1] ). Now using Proposition 2.5.9 we get

E[hu, DGiH] = E[hu, nIn−1(g(·, t))iH] = E[hIn−1(fn−1(·, t)), nIn−1(g(·, t))iH] Z = E[In−1(fn−1(·, t))nIn−1(g(·, t))] dt

[0,1] Z = n(n − 1)! hfn−1, giL2([0,1]n−1) dt [0,1] ˜ = n!hfn−1, giL2([0,1]n = n!hfn−1, giL2([0,1]n) ˜ ˜ = E[In(fn−1)In(g)] = E[In(fn−1)G]

2 n where the third and last lines come from the isometry between Ls([0, 1]) and Hn. One can see from this that δ is an integral. δ(u) is called the Skorokhod stochastic integral of the process u and if the process is adapted then the Skorokhod integral will coincide with the Itô integral, i.e.

1 Z δ(u) = ut dWt. 0 2.7 The generator of the Ornstein-Uhlenbeck semigroup 19

2.7 The generator of the Ornstein-Uhlenbeck semigroup

The Ornstein-Uhlenbeck semigroup is related to the solution of some stochastic differential equations. It is closely related to the gradient and divergence. The generator of this semigroup is denoted L. In this presentation we start with the following deﬁnition.

Deﬁnition 2.7.1. For F ∈ L2(Ω, F,P ) deﬁne

∞ X LF := − nJnF n=0 where Jn denotes the projection to the nth Wiener chaos. We deﬁne Dom L as

( ∞ ) 2 X 2 2 Dom L := F ∈ L : n kJnF k2 < ∞ . n=0

The relation to the gradient and divergence is presented in the next propositions.

Proposition 2.7.2. For F ∈ L2(Ω, F,P ) the following are equivalent: (i) F ∈ Dom L (ii) F ∈ D1,2 and DF ∈ Dom δ. If (i) or (ii) holds then −LF = δDF .

Proof: We know from earlier that for G ∈ D1,2 we have ∞ X E[hDG, DF iH] = nE[JnFJnG]. n=0

1,2 Assume that (ii) holds. Let G ∈ Hn, and F ∈ D , then

E[Gδ(DF )] = E[hDG, DF iH] = nE[JnGJnF ] = nE[GJnF ].

For a general G ∈ D1,2 we get

∞ ∞ ∞ X X X E[Gδ(DF )] = E[hD( JnG),DF iH] = nE[JnGJnF ] = E[G nJnF ]. n=0 n=0 n=0

Since G was arbitrary (i) follows. Assuming (i), and letting G ∈ D1,2 be a test function, we get

∞ ∞ X X E[G · LF ] = − E[GnJnF ] = − nE[JnGJnF ] = −E[hDG, DF iH]. n=0 n=0

By deﬁnition DF ∈ Dom δ and δDF = −LF . 20 2 Chapter 2-Malliavin Calculus

Proposition 2.7.3. We have Z ∈ Dom δ and for F ∈ Z with ∞ n F = f(W (h1),...,W (hn)), f ∈ Cp (R ) we have

n X ∂2 LF = f(W (h ),...,W (h ))hh , h i ∂x ∂x 1 n i j H i,j=1 i j n X ∂ − f(W (h ),...,W (h ))W (h ). ∂x 1 n i i i

1,2 Pn ∂ Proof: Since F ∈ and DF = f(W (h1),...,W (hn))hi we have that D i=1 ∂xi DF ∈ ZH. We recall that for n X 1,2 u := Fjhj,Fj ∈ D , hj ∈ H, j=1 we have n n X X δ(u) = FjW (hj) − hDFj, hji . H j=1 j=1 We now get

n X ∂ δDF = f(W (h ),...,W (h ))W (h ) ∂x 1 n i i=1 i n X ∂ − D f(W (h1),...,W (hn)) , hj ∂xj j=1 H n n n X ∂ X X ∂ = f · W (h ) − h f · h , h i ∂x i ∂x i j H i=1 i j=1 i=1 i n n X ∂ X ∂ = f · W (h ) − f · hh , h i ∂x i ∂x ∂x i j H i=1 i i,j=1 i j and the claim follows. We see that the Malliavin calculus provides the necessary tools for doing analysis on the Wiener space, the most prominent tool being the integration by parts formula. 3 Chapter 3-Dirichlet form theory

IRICHLET forms are a special type of bilinear forms. The theory of Dirichlet forms D was ﬁrst introduced in the works by Beurling and Deny in 1958 and 1959. There is a link between Dirichlet forms and Markov processes. In this chapter the general deﬁnitions are introduced and the links between semigroup, resolvent, generator, and form are stated.

3.1 Introduction

This chapter follows the structure in the book by Z.-M. Ma and M. Röckner [5] and the theory can be found there in a more general form. In this section we will let H be a Hilbert space with corresponding norm k · kH and inner product h·, ·iH.

3.2 Resolvents

The resolvent set of a linear operator L is a set of complex numbers λ for which the operator L − λI is in some sense well-behaved.

Deﬁnition 3.2.1. Let L be a linear operator on H. The resolvent set ρ(L) of L is deﬁned to be the set of all α ∈ R such that (α − L): D(L) → H is one-to-one and for its inverse (α − L)−1 we have

−1 (i) D((α − L) ) = H, −1 (ii) (α − L) is continuous on H. We call {(α − L)−1 : α ∈ ρ(L)} the resolvent of L and σ(L) := R \ ρ(L) the spectrum of L. We see that all eigenvalues are part of the spectrum but the spectrum can have other −1 points as well. For α ∈ ρ(L) we let Ga := (α − L) .

21 22 3 Chapter 3-Dirichlet form theory

Deﬁnition 3.2.2. A family (Gα)α>0) of linear operators on H is called a continuous contraction resolvent on H if:

(i) lim αGαu = u for all u ∈ , (continuity) α→∞ H (ii) Ga is a contraction on H for all α > 0, (iii) Gα − Gβ = (β − α)GαGβ for all α, β > 0. (resolvent equation)

3.3 Semi-groups

In probability theory, semigroups are associated with the transition probabilities of Markov processes.

Deﬁnition 3.3.1. A family (Tt)t>0 of linear operators on H with D(Tt) = H for all t > 0 is called a continuous contraction semigroup on H if

(i) lim Ttu = u for all u ∈ , (continuity) t→0 H (ii) Tt is a contraction on H for all t > 0, (iii) TsTt = Ts+t for all s, t > 0. (semigroup property) It turns out that semigroups are connected to resolvents as will be shown later.

Definition 3.3.2. The infinitesimal generator of a continuous semigroup (Tt)t>0 is defined as the operator (A,D(A)) on H satisfying

Ttu − u D(A) = {u ∈ H : lim exists}, t&0 t

T u − u Au = lim t , u ∈ D(A). t&0 t Given a semigroup and thus also its generator, it is possible to, in a natural way, associate it with a resolvent. The following proposition shows this relation.

Proposition 3.3.3. Let (Tt)t>0 be a continuous contraction semigroup on H with gen- −1 erator A. Then A is densely deﬁned, (0, ∞) ⊂ ρ(A) and if Gα := (α − A) , α > 0 then ∞ Z −αs Gαu = e Tsu ds, u ∈ H, α > 0. 0

We also have that (Gα)α>0 is a continuous contraction resolvent. Proof: The proof of this proposition will be omitted but can be found in [5]. We see that there is a connection between semigroups, resolvents, and the generator A. For an arbitrary linear operator L one might wonder if it is the generator of a semigroup or not. The answer to this quesion is answered by the Hille–Yosida theorem, which characterizes the generators for contraction semigroups. The theorem is named after the mathematicians Einar Hille and Kosaku Yoside who independently discovered the result in 1948. 3.4 Bilinear forms 23

Theorem 3.3.4. (Hille–Yosida) A densely deﬁned linear operator A on H is the generator of a continuous contraction semigroup if any only if (i) (0, ∞) ⊂ ρ(A), (ii) kα(α − A)−1k ≤ 1 for all α > 0. The continuous contraction semigroup is uniquely determined by A and A is closed. Proof: The proof of the Hille–Yosida theorem is quite lengthy and will therefore be omitted. The full proof can be found in [5].

3.4 Bilinear forms

In this section we’ll look into the relationship between the resolvent, semigroup and bilinear form. We will assume that E is positive definite throughout the section. Definition 3.4.1. The symmetric part E˜ of a bilinear form E is defined by 1 E˜(u, v) = (E(u, v) + E(v, u)). 2 Definition 3.4.2. We define

Eα(u, v) = E(u, v) + αhu, viH, u, v ∈ D(E). Following this deﬁnition we can also deﬁne the corresponding norm.

Definition 3.4.3. Given a bilinear form E on a Hilbert space H we define the E1-norm by 1/2 k · kE1 := h·, ·iH + E(·, ·) . Definition 3.4.4. (E,D(E)) satisfies the weak sector condition if there exists a constant K > 0 such that 1/2 1/2 |E1(u, v)| ≤ KE1(u, u) E1(v, v) , ∀u, v ∈ D(E). Definition 3.4.5. A bilinear form (E,D(E)) on H is called coercive if D(E) is a dense linear subspace of H and E : H × H → R such that (i) Its symmetric part (E˜,D(E)) is a symmetric closed form on H. (ii) (E,D(E)) satisfies the weak sector condition. A property of a coercive form is that it grows rapidly as the argument gets bigger. Theorem 3.4.6. Let (E,D(E)) be a coercive closed form on H. Then there exist unique ˜ continuous contraction resolvents (Gα)α>0, (Gα)α>0 on H such that ˜ Gα(H), Gα(H) ⊂ D(E), ˜ Eα(Gαf, u) = hf, uiH = Eα(u, Gαf) for all f ∈ H, u ∈ D(E), α > 0.

G˜α is the adjoint of Gα for all α > 0 and we have ˜ hGαf, giH = hf, Gαgi for all f, g ∈ H. 24 3 Chapter 3-Dirichlet form theory

Proof: The proof of this theorem will be omitted. As in the other cases the full proof can be found in [5].

A consequence of this is the following relation between the bilinear form and the corresponding generator to the resolvent.

Corollary 3.4.7. Let (E,D(E)) and (Gα)α>0 be as above and let (A,D(A)) be the generator of (Gα)α>0 i.e. the unique operator on H such that (0, ∞) ∈ ρ(A) and Gα = (α − A)−1 for all α > 0. Then

D(A) ⊂ D(E),

E(u, v) = h−Au, viH for all u ∈ D(A), v ∈ D(E).

Proof: Let α > 0 and u ∈ D(A). Now u ∈ Gα(H) ⊂ D(E) and we get for all v ∈ D(E)

−1 −1 E(u, v) = Eα(GαGα u, v) − αhu, viH = hGα u, viH − αhu, viH −1 = (Gα − α)u, v = h−Au, vi . H H

Theorem 3.4.8. Let (Gα)α>0 be a continuous contraction resolvent on H with corresponding generator A. Deﬁne

E(u, v) := h−Au, viH, u, v ∈ D(A).

Now (Gα)α>0 and (E,D(E)) have the relation as in the above theorem.

These theorems state the relationship between the resolvent to a linear operator and its corresponding bilinear form.

3.5 Dirichlet forms

The theory of Dirichlet forms is used in the area of Markov process theory. Dirichlet forms are connected to potential theory and energy methods in contrast to partial differential equation theory which are the usual tools used in diffusion theory.

Deﬁnition 3.5.1. A coersive closed form (E,D(E)) on L2(E, m) is called a Dirichlet form if for all u ∈ D(E) it holds

(i) u+ ∧ 1 ∈ D(E), (ii) E(u + u+ ∧ 1, u − u+ ∧ 1) ≥ 0, (iii) E(u − u+ ∧ 1, u + u+ ∧ 1) ≥ 0.

Remark 3.5.2. If E is symmetric a simple calculation shows that conditions (ii) and (iii) above are equivalent to E(u+ ∧ 1, u+ ∧ 1) ≤ E(u, u). 3.5 Dirichlet forms 25

Definition 3.5.3. Let H = L2(E, m). A bilinear form E on L2(E, m) is regular if D(E) ∩ C0(E) is dense in D(E) w.r.t. E1-norm and dense in C0(E) w.r.t. the uniform norm. Here C0(E) denotes all continuous functions on E with compact support. We recall some basic definitions from the theory of stochastic processes. Definition 3.5.4. We say that a function is cadlag if it is right continuous with left limits, i.e.

(i) f(x−) := lim f(t) exists, t%x (ii) f(x+) := lim f(t) exists and is equal to f(x). t&x

The corresponding term for left continuous functions is caglad. Deﬁnition 3.5.5. Two stochastic processes X and Y with a common index set T are called versions of one another if

t ∈ T,P ({ω : X(t, ω) = Y (t, ω)}) = 1.

Such processes are also said to be stochastically equivalent.

If in addition the process Xt is left or right continuous then for a version Yt we have Xt = Yt almost surely. It turns out that we only need to study the form on a dense subset of D(E) and here the concept of nests appear.

Definition 3.5.6. (i) An increasing sequence (Fk)k∈N of closed subsets of E is called an E-nest if ∪ D(E) is dense in D(E) w.r.t. k · k˜ , where D(E) denotes {u ∈ k≥1 Fk E1 Fk D(E): u = 0 m-a.e. on E \ Fk}. c (ii) A subset N ⊂ E is called E-exceptional if N ⊂ ∩k≥0Fk for some E-nest (Fk)k∈N. (iii) We say that a property holds E-quasi-everywhere if it holds everywhere outside some E-exceptional set. We can relate the definition of quasi-continuity to a similar notion on the E-nest. Definition 3.5.7. An E-quasi-everywhere defined function f is called E-quasi-continuous if there exists an E-nest (Fk)k∈N such that f is continuous on (Fk)k∈N. It is now possible to define what we mean with a quasi-regular Dirichlet form. Definition 3.5.8. A Dirichlet form (E,D(E)) on L2(E, m) is called quasi-regular if:

(i) There exists an E-nest (Fk)k∈N consisting of compact sets, ˜1/2 (ii) There exists an E1 -dense subset of D(E) whose elements have E-quasi-continuous m-versions,

(iii) There exist un ∈ D(E), n ∈ N, having E-quasi-continuous m-versions u˜n, n ∈ N, and an E-exeptional set N ⊂ E such that {u˜n : n ∈ N} separates the points of E \ N. 26 3 Chapter 3-Dirichlet form theory

In inﬁnite dimension the quasi-regularity of the Dirichlet form is enough to be able to associate it with a special kind of process. The following deﬁnitions serve to clarify what we mean.

Deﬁnition 3.5.9. A process M with state space E is called µ-tight if there exists an increasing sequence (Kn)n∈N of compact sets in E such that

Pµ( lim inf{t > 0 : Mt ∈ E \ Kn} < ∞) = 0 n→∞ where inf ∅ := ∞. We note that it simply means that for every ε > 0 there exists a compact set K such that Pµ(Mt ∈ K) > 1 − ε. Deﬁnition 3.5.10. A cadlag Markov process M with state space E and transition semigroup (pt)t>0 is said to be properly associated with (E,D(E)) and its corresponding semigroup (Tt)t>0 if ptf is an E-quasi continuous µ-version of Ttf for all t > 0 and all bounded f ∈ L2(E; µ).

The following theorem is the connection between the theory of forms and process theory.

Theorem 3.5.11. Let (E,D(E)) be a quasi-regular Dirichlet form on L2(E, µ). Then there exists a pair (M, Mˆ ) of µ-tight special standard processes which is properly associated with (E,D(E)).

Proof: The proof is lengthy and will therefore be omitted. The reader may ﬁnd it in [5].

Remark 3.5.12. It is known that if E is locally compact then there exists an associated process to every regular Dirichlet form on L2(E, µ). See [3].

The Cameron–Martin formula tells us how the Wiener measure changes under a trans- lation. It was discovered by Robert Cameron and William Martin in 1944.

Theorem 3.5.13. Let h ∈ H, where

1 Z 0 2 H = {f ∈ C([0, 1]) : f(0) = 0, f absolutely continuous, f (u) du < ∞}, 0 and deﬁne Th(z) := z + h, z ∈ E, where

E = {f ∈ C([0, 1]) : f(0) = 0}.

−1 Let µ ◦ Th denote the image measure of µ under Th. Then

dµ ◦ T −1 1 h = exp W − khk2 . dµ h 2 H 3.5 Dirichlet forms 27

Proof: Let f, h ∈ H and α := hf, hiH/hh, hiH. Then Wh and Wf−αh are independent since they are Gaussian and

E[WhWf−αh] = hh, f − αhiH = hh, fiH − hf, hiH = 0. We have Z 1 exp {ihf, zi } exp W (z) − khk2 µ(dz) H h 2 H 1 Z = exp − khk2 exp {ihf, zi + hh, zi } dµ 2 H H H 1 Z = exp − khk2 exp {ihf − αh, zi + h(1 + iα)h, zi } dµ 2 H H H 1 Z = exp − khk2 exp {iW + (1 + iα)W } dµ 2 H f−αh h 1 Z Z = exp − khk2 exp{iW } dµ exp{i(−i + α)W } dµ 2 H f−αh h 1 1 1 = exp − khk2 exp{− kf − αhk2 } exp{− (−i + α)2khk2 } 2 H 2 H 2 H 1 1 1 1 = exp − khk2 − kf − αhk2 + khk2 + iαkhk2 − α2khk2 2 H 2 H 2 H H 2 H 1 = exp − hf, fi + ihf, hi 2 H H Z −1 = exp {ihf, ziH} (µ ◦ Th )(dz).

Keeping all these deﬁnitions and theorems in mind we are now well equipped to study the Dirichlet form in the next chapter.

4 Chapter 4-Main results

HE paper included here is a joint work with Jörg-Uwe Löbus and is the result of a T study at the department of Mathematical Statistics at Linköping University during 2012. This version of the paper contains slightly more details than the version available at ArXiv and the version submitted for publication. The aim of the paper is to take results from the ﬁnite dimensional theory into the inﬁnite dimensional case. This is done via the means of a coordinate representation.

4.1 Earlier work

In the paper Construction of diffusions on path and loop spaces of compact Riemannian manifolds [2] the idea of Dirichlet form theory is used to create a diffusion process. More precisely the idea is that every quasi-regular Dirichlet form has an associated diffusion process. In [2] a form deﬁned on a compact Riemannian manifold with these properties is presented. The paper A class of processes on the path space over a compact riemannian manifold with unbounded diffusion [4] expands on the same idea. Here the form is Z hDF, DAGiH dν (4.1) where the operator A is deﬁned by

∞ X AΦ(γ) := λihSi, Φ(γ)iHSi, Φ ∈ D(A), i=1 and the diffusion coefﬁcients λ are assumed to be bounded from below and possibly 1−ε unbounded. It is shown that under the assumption λn ≤ cn , the form is a quasi- regular Dirichlet form and thus has an associated process.

29 30 4 Chapter 4-Main results

The paper presented in this thesis is similar to the paper Quasi-regular Dirichlet forms on Riemannian path and loop spaces [11]. Both these papers investigate infinite dimensional diffusion processes where the diffusion is related to possibly unbounded operators. Both papers refer to the ideas of [4]. Let us summarize the differences between this paper and [11]. The main difference is that the theory in [11] is presented in a geometric setting, i.e. on a manifold while here we study the flat case. Even though the geometric framework in [11] is more general, our coordinatewise non-geometric setting makes it easier to focus on the phenomenon we are interested in to investigate, namely pointwise unboundedness of diffusion. The forms studied in these two papers are of type (4.1). The form in [11] differs from the one presented here in the sense that the operator A in this paper is defined pointwise, constant over the trajectory space, while in [11] it can vary, i.e. A = A(γ). The operator in [11] is defined by letting (A, D(A)) be a densely defined self-adjoint operator 2 on L (W0 → H; µ) such that ∞ 2 AfΦ = fAΦ for f ∈ L (µ) and Φ ∈ L (W0 → H; µ), (A0) and A ≥ εI for some ε > 0.

Here W0 is a manifold and I denotes the identity operator. They deﬁne Z 1/2 1/2 EA(F,G) = hA DF,A DGiH dµ, F, g ∈ Y,

W0 where ∞ 1/2 Y := {F ∈ FCb : DF ∈ D(A )}, ∞ and FCb denotes the set of cylindrical functions {F (γ) = f(γ(s1), . . . , γ(sk)) : sj ∈ ∞ [0, 1], f ∈ Cb }. In addition the assumption

1/2 2 2 ∞ d kA Φt,vk 2 ≤ Ckvk ∞ d , t ∈ [0,T ], v ∈ L (W0 → R ; µ), L (W0→H;µ) L (W0→R ;µ) (A1) is made. In our situation with A constant on the trajectory space and with non-decreasing eigenvalues, the conditions (A0) and (A1) are satisﬁed. In [11] the condition

d ∞ d X X X −m µ(λi) + max µ(λd(2m+k−1)+j)2 < ∞ 1≤k≤2m j=1 m=0 j=1 is shown to be sufﬁcient for the closability of the form. In our case the condition is reduced to ∞ X λd2p < ∞, 2p p=0 a relation ﬁrst noted in [4]. Moreover we show that this condition is necessary in the sense of Proposition 4.5.6 (c). In order to handle weights with respect to the Wiener measure [11] introduces geometric conditions (1.3), (1.5), and (1.6). We use the condition (4.12) in Lemma 4.5.3 which is compatible with [2]. To show quasi-regularity both this paper and [11] use the method of [2], [4], and [9]. However, the proof in [11] involves a (geometric) result of [10]. In contrast, our proof uses Doob’s inequality in that particular point of the proof. 4.2 Introduction 31

4.2 Introduction

This paper is concerned with Dirichlet forms of type * ∞ + Z Z X E(F,G) = hDF, ADGi ϕdν = DF, λi hSi,DGi Si ϕdν, (4.2) H H i=1 H where the diffusion operator A is in general unbounded, cf. [4]. We are interested in weight functions ϕ of the form  1 1  Z Z  1 2  ϕ(γ) = exp hb(γ ), dγ i d − kb(γ )k ds . s s R s  2  0 0 This choice of weight functions is motivated by the papers [11, 12] by F.-Y. Wang and B. Wu. In this way it is possible to relate and compare the results of the present paper to the findings there. We present our ideas in terms of infinite dimensional processes on the classical Wiener space using a coordinate representation. This makes the subject comprehensible, in particular, to readers familiar with the finite dimensional theory. The form is studied on the set of smooth cylindrical functions

F,G ∈ Y = {F (γ) = f(γ(s1), . . . , γ(sk)) : sj is a dyadic point}, where γ is a Wiener trajectory. We do this as well on the more common set of cylindrical functions F,G ∈ Z = {F (γ) = f(γ(s1), . . . , γ(sk)) : sj ∈ [0, 1]}. Well-definiteness of E on Y is a consequence of the fact that the sum in (4.2) is finite. Well-definiteness of E on Z requires the convergence of the sum in (4.2). These two different initial situations result in possibly different closures of (E,Y ) and (E,Z) on L2(ϕν). Using the coordinate representation in (4.2) we give conditions for closability. The requirement of ϕ−1 ∈ L1(ν) would be sufficient, for example, for closability in the 2 classical case with bounded cylindrical functions and λ1 = λ2 = ... = 1 on L (ν), see [5]. However, the paper investigates forms of the structure (4.2) with an in general unbounded diffusion operator. We give necessary and sufficient conditions on the increase 2 of λ1, λ2,... that guarantee closability of (E,Y ) and (E,Z) on L (ϕν) in terms of the Schauder functions Si, i ∈ N, the coordinate functions in H. Locality, Dirichlet property, and quasi-regularity of the closure (E,Z) on L2(ϕν) is then obtained by using methods of [1, 2, 4, 5, 9]. We are also interested in characterizing the associated process in terms of their local first and second moments of the form 1 Z lim (Si(γ) − Si(τ)) Pτ (Xt ∈ dγ), (4.3) t→0 t and Z 1 2 lim (Si(γ) − Si(τ)) Pτ (Xt ∈ dγ), (4.4) t→0 t where Si are certain linear functions on the trajectory space. We represent these local moments in terms of the sequence λ1, λ2,... and the weight function ϕ. Corresponding to the closability condition of (E,Z) on L2(ϕν), we derive a necessary and sufficient condition for the limits (4.3) and (4.4) to exist. This way lets us obtain compatibility with the classical Kolmogorov characterization of finite dimensional diffusion processes. 32 4 Chapter 4-Main results

4.3 Formal deﬁnitions

2 2 d We study the form on the space L (ϕν) ≡ L (Ω, ϕν) where Ω := C0([0, 1]; R ) := {f ∈ C([0, 1]; Rd, f(0) = 0}, ν is the Wiener measure on Ω and ϕ is a density function speciﬁed below. As stated earlier, the form is given by Z E(F,G) = hDF, ADGi ϕdν, F, G ∈ D(E), (4.5) H where H is the Cameron-Martin space, i.e., the space of all absolutely continuous Rd- valued functions f on [0, 1], with f(0) = 0 and equipped with inner product Z 0 0 hϕ, ψi := hϕ (x), ψ (x)i d dx. H R [0,1]

Motivated by [11], we suppose in sections 4.7 and 4.8 that ϕ :Ω → [0, ∞] has the form

 1 1  Z Z  1 2  ϕ(γ) = exp hb(γ ), dγ i d − kb(γ )k d ds s s R s (4.6)  2 R  0 0

d d 2 d where b : R → R is the gradient of a function f ∈ Cb (R ; R). Then by Itô’s formula, ϕ deﬁned by (4.6) is both bounded from below and from above, on Ω. We deﬁne the set of all cylindrical functions n Z := F (γ) = f (γ(s1), . . . , γ(sk)) , γ ∈ Ω:

∞ dk o 0 < s1 < ··· < sk = 1, f ∈ Cp (R ), k ∈ N

∞ where Cp denotes smooth functions with polynomial growth. We also deﬁne n Y := F (γ) = f (γ(s1), . . . , γ(sk)) , γ ∈ Ω:

l n o F ∈ Z, s1, . . . , sk ∈ 2n : l ∈ {1,..., 2 } , n ∈ N . For F ∈ Z and γ ∈ Ω the gradient operator D is deﬁned by

k X DsF (γ) = (si ∧ s)(∇si f)(γ), s ∈ [0, 1], (4.7) i=1 where (∇si f)(γ) = (∇si f)(γ(s1), . . . , γ(sk)) denotes the gradient of the function f relative to the ith variable while holding the other variables ﬁxed. We let (ej)j=1,...,d denote the standard basis in Rd and

H1(t) = 1, t ∈ [0, 1],

 m/2 k−1 2k−1 2 if t ∈ 2m , 2m+1  2k−1 k m m/2 m H2 +k(t) = −2 if t ∈ 2m+1 , 2m k = 1,..., 2 , m = 0, 1,...,  0 otherwise 4.4 Deﬁnition of the form 33 denote the system of the Haar functions on [0, 1]. We also deﬁne

gd(r−1)+j := Hr · ej, r ∈ N, j ∈ {1, . . . , d}, (4.8) and s Z Sn(s) := gn(u) du, s ∈ [0, 1], n ∈ N. 0

4.4 Deﬁnition of the form

We use the following deﬁnitions from [4]. We choose a non-decreasing sequence of positive numbers λ1, λ2,... .

( Z ∞ ) 2 X 2 2 D(A) := Φ ∈ L (Ω → , ν): λ hSi, Φi dν < ∞ , H i H i=1

∞ X AΦ(γ) := λi hSi, Φ(γ)i Si, γ ∈ Ω, Φ ∈ D(A). H i=1 We can then conclude that

* ∞ + Z Z X E(F,F ) = hDF, ADF i ϕdν = DF, λi hSi,DF i Si ϕdν H H i=1 H ∞ Z X 2 = λi hSi,DF i ϕdν < ∞,F ∈ Y, (4.9) H i=1 is well deﬁned since, for F ∈ Y , this is just a ﬁnite sum.

4.5 Closability

In this section we ﬁrst prove a general closability result for the Malliavin gradient. Using this result, a criterion for the closability of the form is formulated. We note that, under these conditions, the form is local. We assume that ϕ ≤ c < ∞.

2 2 Lemma 4.5.1. If xn → x in L (ϕν; H) where H = L (T ), T = [0, 1], and Z lim ψδ(xn) dν = 0 (4.10) n→∞ for all ψ in Z, then x = 0.

2 m+1 Proof: For any φ ∈ Hm+1, we have φ = Im+1(g) for some symmetric g ∈ L (T ). We have for all n ∈ N ∞ X (n) xn(t) = Ij fj (·, t) , (4.11) j=0 34 4 Chapter 4-Main results

(n) 2 j+1 (n) where fj ∈ L (T ) where for j ≥ 1, fj is symmetric in the ﬁrst j variables. Thus

∞ X ˜(n) δ(xn) = Ij+1(fj ). j=0

We can restrict ourselves to φ ∈ Hm+1 such that φ ∈ Z. By hypothesis we have Z ˜(n) (n) (m + 1)! g, fm L2(T m+1) = Im+1(g)Im+1(fm ) dν Z = φδ(xn) dν → 0, as n → ∞.

Since g ∈ L2(T m+1) is symmetric we have

(n) g, fm L2(T m+1) → 0, as n → ∞.

For all symmetric h ∈ L2(T m, H) we obtain

(n) 2 m hh, fm (·, t)iL (T ;H) → 0, as n → ∞, where the letter t indicates the variable for the function in H. Using the fact that the mth 2 m Wiener chaos is isometric to the space of symmetric functions Ls(T ) we get for any 2 ψ ∈ L (ν; H) such that ψ(·, t) ∈ Hm for a.e. t ∈ [0, 1], ψ = Im(h), we have

D (n) E Im(h),Im f (·, t) → 0, as n → ∞. m 2 L (ν;H)

(n) Since Im fm (·, t) is the projection of xn to the mth Wiener chaos we have

˜ hxn, φiL2(ν; ) −→ 0, H n→∞

˜ 2 for all ﬁnite linear combinations φ = α1ψ1 + ... + αkψk, where ψi ∈ L (ν; H), ψi(·, t) is from the ith Wiener chaos for a.e. t ∈ [0, 1], and i ∈ {1, . . . , k}. Since ϕ is bounded this implies ˜ hxn, φiL2(ϕν; ) −→ 0. H n→∞

2 Since also kxnkL (ϕν;H) is bounded we have ˜ ˜ 2 hxn, φiL2(ϕν; ) −→ 0, for all φ ∈ L (ϕν; ). H n→∞ H

2 This contradicts xn → x in L (ϕν; H) and the statement follows.

Lemma 4.5.2. If f ∈ L1(ν), ∀f ∈ Z, (4.12) ϕ 1 2 then { ϕ Dψ : ψ ∈ Z} is a dense subset of L (ϕν; H). 4.5 Closability 35

Proof: We assume the contrary. Then we can ﬁnd x ∈ L2(ϕν; H) and cylindrical func- 2 tions xn, n ∈ , with values in , such that x 6= 0 and xn −→ x in L (ϕν; ) and N H n→∞ H

Z 1 hDψ, xi ϕ dν = 0 ϕ H for all ψ ∈ Z. It follows that

Z 1 Z Z 0 = hDψ, xi ϕdν = lim hDψ, xni dν = lim ψδ(xn) dν ϕ H n→∞ H n→∞ for all ψ ∈ Z. From this we get x = 0 by Lemma 4.5.1, and we have a contradiction.

Lemma 4.5.3. If 0 < ϕ ≤ c for some c ∈ R+, and (4.12) holds, then (D,Z) deﬁned by 1 Z D(f, g) := hDf, Dgi ϕdν, f, g ∈ Z, 2 H is closable on L2(ϕν). Let L denote the generator of D and note that L is the Ornstein- Uhlenbeck operator.

Proof: Let un ∈ Z, ψ ∈ Z and

2 2 un −→ 0 in L (ϕν), Dun −→ f in L (ϕν; ). n→∞ n→∞ H

Then 1 Z 1 1 Z Du , Dψ ϕdν = hDu , Dψi dν 2 n ϕ 2 n H H Z = un(−Lψ) dν Z 1 = u (−Lψ) ϕdν. n ϕ

By assumption we have 1/ϕ · (−Lψ)2 ∈ L1(ν), i.e. 1/ϕ · (−Lψ) ∈ L2(ϕν). Thus

Z 1 un (−Lψ) ϕdν −→ 0. ϕ n→∞

1 2 Since Lemma 4.5.2 showed that { ϕ Dψ : ψ ∈ Z} is a dense subset of L (ϕν; H), it now follows that f = 0. Hence the form is closable.

Remark 4.5.4. The closability condition (4.12) would weaken to 1/ϕ ∈ L1(ν) if the initial deﬁnition of D was on all bounded f, g ∈ Z. We get compability with [5], section II.2 a).

We also formulate the following lemma that serves an important role in proving and formulating the closability conditions of the form. 36 4 Chapter 4-Main results

Pr −i p−1 Pp−1 p−q−1 Lemma 4.5.5. (a) If sk = i=1 ci · 2 , ip,j = 2 (d + j − 1) + 1 + q=0 cq2 and j ∈ {1, . . . , d} where c1, . . . , cr−1 ∈ {0, 1}, cr = 1, c0 = cr+1 = 0 and r ≥ 2, then

∞ r r+1 !2 X 2 2 X p−1 −p cp X −q λihSi(sk), eji d = λjs + λi 2 cp2 + (−1) 2 cq . R k p,j i=1 p=1 q=p+1

(b) The relation

r r+1 !2 X p−1 −p cp X −q sup λip,j 2 cp2 + (−1) 2 cq < ∞ c1,...,cr−1∈{0,1}, p=1 q=p+1 cr =1,cr+1=0,j∈{1,...,d},r≥2 (4.13) is equivalent to ∞ X λd2p < ∞. (4.14) 2p p=0

r Proof: a) Since ej ⊥ Si for i mod d 6= j and Si(sk) = 0 for i > d2 we get

∞ 2r X 2 2 X 2 λi(Si(sk) · ej) = λj(Sj(sk) · ej) + λdi+j(Sdi+j(sk) · ej) . i=1 i=1

Now since 2 2 λj(Sj(sk) · ej) = λjsk, and Sdi+j(sk) = 0

p−1 Pp−1 p−q−1 unless di + j = 2 (d + j − 1) + 1 + q=0 cq2 for some p ≥ 1. We get

2r X 2 λdi+j(Sdi+j(sk)) · ej) i=1 r X 2 = λip,j (Sip,j (sk) · ej) , p=1 where

( p−1 r+1 2 P −q 2 q=p+1 2 cq if cp = 0 Sip,j (sk) · ej = p+1 p−1 r+1 − 2 2 P −q 2 − 2 q=p+1 2 cq if cp = 1 r+1 p−1 −p cp X −q = 2 2 cp2 + (−1) 2 cq , q=p+1 and thus the statement follows.

p−1 p (b) Assuming (4.13). Recall λ1, λ2,... is nondecreasing and thus λd2 < λip,j ≤ λd2 . 4.5 Closability 37

Using c1 = c3 = c5 = ··· = 1, c2 = c4 = c6 = ··· = 0 we get

r r+1 !2 X p−1 −p cp X −q sup λip,j 2 cp2 + (−1) 2 cq c1,...,cr−1∈{0,1}, p=1 q=p+1 cr =1,cr+1=0,r≥2 ∞ ∞ X 2 1 X λip,j ≥ λ 2p−1 2−p−1 = ip,j 8 2p p=1 p=1 ∞ ∞ 1 X λd2p−1 1 X λd2p ≥ = . 8 2p 16 2p p=1 p=0

Assuming (4.14) we obtain

r r+1 !2 X p−1 −p cp X −q sup λip,j 2 cp2 + (−1) 2 cq c1,...,cr−1∈{0,1}, p=1 q=p+1 cr =1,cr+1=0,r≥2 ∞ ∞ ∞ X 2 1 X λip,j 1 X λd2p ≤ λ 2p−1 2−p = ≤ . ip,j 2 2p 2 2p p=1 p=1 p=1

The statement now follows.

Proposition 4.5.6. Let ϕ satisfy (4.12). 2 (a) The form (E,Y ) is closable in L (ϕν). Let (E,DY (E)) denote the closure of (E,Y ) in L2(ϕν). (b) If ∞ X λd2p < ∞, (4.15) 2p p=0 2 then (E,Z) is closable and we let (E,DZ (E)) denote the closure of (E,Z) on L (ϕν). Furthermore DZ (E) = DY (E) under (4.15). (c) Z ⊂ DY (E) if and only if (4.15). Proof: (a) We have ∞ Z Z X 2 D 1/2 1/2 E E(F,F ) = λi hSi,DF i ϕdν = A DF,A DF ϕdν. H i=1 H

2 Suppose {Fn}n≥1 ∈ Y such that Fn −→ 0 in L (ϕν) and E(Fn − Fm,Fn − Fm) → 0. n→∞ 1/2 2 Since it follows that A DFn is Cauchy in L (Ω → H, ϕν) we may deﬁne

1/2 ψ := lim A DFn. n→∞ We also deﬁne ∞ X −1/2 Jh := λ hSi, hi Si. (4.16) i H i=1 38 4 Chapter 4-Main results

The operator J is bounded in L2(Ω → H, ϕν) and it follows 1/2 2 DFn = JA DFn −→ Jψ in L (Ω → , ϕν). n→∞ H From Lemma 4.5.3, it is known that (D,Z) is closable on L2(ϕν). It follows that −1/2 DFn −→ 0 and thus Jψ = 0. Since λi > 0 and λ > 0, (4.16) gives ψ = 0. n→∞ i 1/2 R 1/2 1/2 Now A DFn −→ 0 and thus E(Fn,Fn) = A DFn,A DFn ϕdν → 0 as n→∞ H n → ∞. (b) We show that (4.15) implies Z ⊂ DY (E). Then Y ⊂ Z ⊂ DY (E). Since (E,Y ) is closable with closure (E,DY (E)), cf. (a), (E,Z) is then also closable and has (E,DY (E)) as its closure i.e. DY (E) = DZ (E). Let xv(p) denote the vth coordinate of p ∈ Rd, v ∈ {1, . . . , d}. Let us demonstrate that v F (γ) = x (γ(s)) ∈ DY (E) for all v ∈ {1, . . . , d} and all s ∈ [0, 1]. Fix s ∈ [0, 1], v ∈ {1, . . . , d}, and let sk −→ s where sk is a sequence of dyadic numbers. Now let k→∞ v Fv,k(γ) := x (γ(sk)) ∈ Y ⊂ DY (E), k ∈ N. Using Lemma 4.5.5 we have ∞ ∞ ∞ X 2 X 2 X 2 λihSi,DF (γ)i = λihSi(s), evi d ≤ sup λihSi(sk), evi d H R R i=1 i=1 k i=1 ∞ ∞ 1 X λip,v X λd2p ≤ λ + ≤ < ∞ (4.17) v 2 2p 2p p=1 p=0

2 where ip,v is constructed as in Lemma 4.5.5. Furthermore Fv,k −→ F in L (ϕν) and k→∞ ∞ ∞ X X λ hS ,D(F − F )(γ)i2 ≤ λ hS ,DF (γ)i2 i i v,k H i i H i=1 i=nk for some sequence nk → ∞ as k → ∞. This says Fv,k −→ F in E1-norm and F ∈ k→∞ 2 2 D (E), where we recall that k · k = E(·, ·) + k · k 2 . Indeed, the same conclusion Y E1 L (ϕν) applies to an arbitrary F ∈ Z with F (γ) = f (γ(s1), . . . , γ(sk)), s1, . . . , sk ∈ [0, 1] since because of (4.7) we have

d k X X ∂f hS ,DF (γ)i = (γ(s ), . . . , γ(s )) hS (s 0 ), e i d . i H 1 k i i j R ∂xi0,j j=1 i0=1 We then apply (4.9) and (4.17). (c) Recalling (b), we still have to show that Z ⊂ DY (E) implies (4.15). We suppose Z ⊂ DY (E). Therefore xv(γ(s)) ∈ DY (E) for all v ∈ {1, . . . , d} and all s ∈ [0, 1]. First we check the expression of Lemma 4.5.5 for all dyadic points sk. To get a bound on ∞ X 2 λihSi(s), eji d , R i=1 when s ∈ [0, 1] is no longer dyadic, we need (4.15). The statement follows. 4.6 Quasi-regularity 39

Proposition 4.5.7. Let ϕ satisfy (4.12). The form (E,DZ (E)) is a Dirichlet form on L2(ϕν).

Proof: We use Proposition I.4.10 from [5]. It follows that we must show E(1 ∧ F +, 1 ∧ F +) ≤ E(F,F ). We know that for F ∈ Y

∞ Z ∞ Z X 2 X 2 E(F,F ) = λi hSi,DF i ϕdν = λi (∂S ,F ) ϕdν H i i=1 i=1 ∞ Z 2 X d = λi F (γ + tSi) ϕdν. dt i=1 t=0

Let ξε : R → [−ε, 1 + ε] be non-decreasing such that ξε(t) = t for all t ∈ [0, 1], 0 + 0 ≤ ξε ≤ 1. It follows that ξε ◦ F → 1 ∧ F . An application of the chain rule gives Z 2 d E(ξε ◦ F, ξε ◦ F ) = ξε ◦ F (γ + tS) ϕdν dt t=0 Z 2 0 d = ξε(F (γ)) · F (γ + tS) ϕdν dt t=0 Z 2 d ≤ 1 · F (γ + tS) ϕdν dt t=0 = E(F,F ), from which we derive E(1 ∧ F +, 1 ∧ F +) ≤ E(F,F ). Thus E is a Dirichlet form.

Proposition 4.5.8. Let ϕ satisfy (4.12). The form (E,DZ (E)) is local.

Proof: This is shown in the same manner as in Proposition 3.4 of [4] using the fact that ν-a.e. implies ϕν-a.e.

4.6 Quasi-regularity

It turns out that the closability condition found in the previous sections is sufﬁcient for the form to be quasi-regular. Throughout this section we assume (4.12).

Proposition 4.6.1. Suppose (4.15). The form described by the closure of

∞ Z X 2 E(F,F ) = λi hSi,DF i ϕdν, F ∈ Z, H i=1 in L2(ϕν), is quasi-regular.

Proof: The conditions for closability can be found in Proposition 4.5.6. We follow the procedure of [4], see also [2, 9]. r−1 r−1 −r Step 1: For r ∈ N, l ∈ {0,..., 2 − 1}, and k = 2 + l, set sk := (2l + 1)2 . Let 40 4 Chapter 4-Main results xv(p) denote the vth coordinate of p ∈ Rd, v ∈ {1, . . . , d}. Fix τ ∈ Ω, k = 2r−1 + l, v v d and v ∈ {1, . . . , d}. Consider the functions fv,k,τ (p) := x (p) − x (τ(sk)), p ∈ R , and v v Fv,k,τ (γ) := fv,k,τ (γ(sk)) = x (γ(sk)) − x (τ(sk)), γ ∈ Ω; clearly Fv,k,τ ∈ Y . Using the procedure of Proposition 4.5.6(b) we get

E(Fv,k,τ ,Fv,k,τ ) ≤ C1 < ∞ (4.18) where C1 the bound obtained from (4.17). Step 2: Let Gn,τ = sup |Fv,k,τ |, n ∈ N, k∈{1,...,n} v∈{1,...,d} then Z ∞ X 2 E(Gn,τ ,Gn,τ ) ≤ sup λi hSi,DFv,k,τ (γ)i ϕdν ≤ C1 < ∞, H k∈{1,...,n} i=1 v∈{1,...,d}

2 as in [4] Lemma 3.2. We show Gn,τ ∈ L (ν).  2 2 v v E Gn,τ ≤ E  sup |x (γ(s)) − x (τ(s))|   v∈{1,...,d}   s∈[0,1]   " d #  v 2 X v 2 ≤ 4E  sup (x (γ(s)))  ≤ 4E sup (x (γ(s))) v∈{1,...,d} v=1 s∈[0,1] s∈[0,1] " # 1 2 = 4dE sup x (γ(s)) ≤ 16d =: C2 < ∞, s∈[0,1] where the last line is obtained using Doob’s inequality. Thus

E1(Gn,τ ,Gn,τ ) ≤ C1 + C2 =: C3 < ∞. (4.19)

Step 3: Using (4.19) and the fact that the sequence (Gn,τ )n∈N satisﬁes Gn,τ ≤ Gn+1,τ , n ∈ N. By using Lemma I.2.12 of [5] we get that the function v v Hτ (γ) := sup x (γ(s)) − x (τ(s)) , γ ∈ Ω, s∈[0,1] v∈{1,...,d} belongs to D(E) and that E1(Hτ ,Hτ ) ≤ C3. (4.20)

Let {τk : k ∈ N} be a dense set in Ω. Set

Kn := inf Hτk , n ∈ N. (4.21) 1≤k≤n 4.7 The generator 41

1/2 Again using Lemma I.2.12 of [5] we have Kn ∈ D(E) and Kn bounded in (D(E), E1 ) from (4.20). We apply the Banach-Saks theorem, which states that every bounded se- 1/2 quence in (D(E), E1 ), has a subsequence whose Cesaro means converge strongly and we get E1(Kn,Kn) −→ 0 since {τk} is dense. As Kn is continuous, we may use Propo- n→∞ sition 3.5 of [5] Chapter III, from which it follows that there exists a subsequence Knk , k ∈ N, and an E-nest Fm, m ∈ N, such that Knk converges uniformly to zero on each Fm as k → ∞. Let us follow an idea of [9], proof of Proposition 3.1. Given δ > 0 we can ﬁnd k such that Knk < δ. We have

n [k Fm ⊂ B(τi, δ), i=1 by (4.21), where B(x, δ) denotes the ball of radius δ centered at x. Now it follows that each Fm is totally bounded. Fm closed and totally bounded implies Fm compact. Thus Fm, m ∈ N forms an E-nest consisting of compact sets. Step 4: For ﬁxed τ ∈ Ω, the system of functions Fv,k,τ , v ∈ {1, . . . , d}, k ∈ N separates the points in Ω, Step 3 showed that there is an E-nest consisting of compact sets and the form is a Dirichlet form by Proposition 4.5.8. Quasi-regularity now follows from its deﬁnition. Remark 4.6.2. We observe that to obtain (4.18) we need the condition (4.15). We recall that by Proposition 4.5.6(b) we have (E,DY (E)) = (E,DZ (E)).

Corollary 4.6.3. There exists a diffusion process M associated with (E,DZ (E)). Proof: The result is an immediate consequence of Propositions 4.5.8 and 4.6.1 using Theorem IV.3.5 of [5].

4.7 The generator

Let ϕ(γ) have the form of (4.6) and (E,DY (E)) given by

∞ Z X Z E(F,G) = hDF, ADGi ϕdν = λi ∂Si F ∂Si G ϕdν, F, G ∈ Y i=1

2 where DY (E) is the closure of (E,Y ) on L (ϕν). We determine the generator A of this form, i.e. Z E(F,G) = (−AF )G ϕdν, F ∈ D(A),G ∈ DY (E).

R 1 g S (γ) := hg (s), dγ i d i ∈ S (γ), S (γ),... Using n from (4.8), let i 0 i s R , N . We observe that 1 2 are independent N(0, 1). Proposition 4.7.1. We have Y ⊂ D(A). If F ∈ Y then ∞ X 2 ∂Si ϕ∂Si F AF = λi ∂ F + + Si∂S F . (4.22) Si ϕ i i=1 42 4 Chapter 4-Main results

Proof: Let F,G ∈ Y . Once again we note that the following sum is just a ﬁnite sum because of the particular structure of Y . We have

∞ X Z E(F,G) = λi ∂Si F ∂Si G ϕdν i=1 ∞ X 1 Z Z = lim λi G(γ + sSi)∂S F ϕdν − G∂S F ϕdν s→0 s i i i=1 ∞ X 1 Z Z = lim λi G∂S F (γ − sSi)ϕ(γ − sSi) dν(γ + sSi) − G∂S F ϕdν s→0 s i i i=1 ∞ " Z X dν ◦ TsSi ∂Si F (γ − sSi) − ∂Si F = lim λi Gϕ(γ − sSi) · dν s→0 dν s i=1 Z dν ◦ T ϕ(γ − sS ) − ϕ + G sSi ∂ F · i dν dν Si s dν◦T ! # Z sSi − 1 + G∂ F ϕ dν dν Si s ∞ Z X 2 d dν ◦ TsSi = λi −Gϕ∂ F − G∂Si F ∂Si ϕ + Gϕ∂Si F dν, Si ds dν i=1 s=0

where T is the shift operator, TsSi (γ) = γ + sSi. Thus the generator satisﬁes

∞ X 2 ∂Si ϕ ∂Si F d dν ◦ TsSi AF = λi ∂ F + − ∂Si F Si ϕ ds dν i=1 s=0 from which we can conclude F ∈ D(A). The Cameron-Martin formula gives

d dν ◦ TsSi (γ) = −Si(γ) ds s=0 dν and (4.22) follows immediately.

b hb, e i d g Remark 4.7.2. Let j denote j R and let i be as in (4.8). The particular structure of ϕ gives

1 1 d Z Z ∂Si ϕ X = hDb (γ ),S (s)i d(γ ) + hb(γ ), g (s)i d ds ϕ j s i H j s s i R j=1 0 0 1 Z 1 2 − hDkb(γs)k d ,Sii ds 2 R H 0 4.8 Local ﬁrst moment 43 and therefore

1 ∞ " d Z X 2 X AF = λ ∂ F + S ∂ F + h∇b (γ ),S (s)i d d(γ ) i Si i Si j s i R j s i=1 j=1 0 1 1 * d + # Z Z X + hb(γ ), g (s)i d ds − b (γ )∇b (γ ),S (s) ds ∂ F . s i R j s j s i Si j=1 d 0 0 R

4.8 Local first moment Let M = (Xt)t∈[0,1], (Pγ )γ∈Ω be the corresponding diffusion process defined in Corol- v S hS , e i d S (γ) (A,D(A)) lary 4.6.3, i denote i v R , and i be as in Chapter 4.7. We recall that 1/2 2 2 is the closure of (A,Y ) in the graph norm k · kL2(ϕν) + kA · kL2(ϕν) . Thus Si ∈ Y ⊂ D(A) by definition.

Proposition 4.8.1. We have Z 1 ∂Si ϕ(τ) lim (Si(γ) − Si(τ)) Pτ (Xt ∈ dγ) = λi + Si(τ) . t→0 t ϕ(τ)

Proof: According to Proposition (4.22), we have Si(γ) ∈ Y ⊂ D(A). Therefore

1 Z lim (Si(γ) − Si(τ)) Pτ (Xt ∈ dγ) = A(Si(τ)) t→0 t and by (4.22)

∞ X 2 ∂Sj ϕ∂Sj Si(τ) A(Si(τ)) = λj ∂ Si(τ) + + Sj(τ)∂S Si(τ) Sj ϕ j j=1 ∂ ϕ(τ) = λ Si + S (τ) . i ϕ(τ) i

Pr −i For a dyadic number, sk = i=1 ci · 2 , we note that there exist N(sk) ∈ N such that h(sk ∧ ·),SiiH = 0 whenever i > N(sk).

v Proposition 4.8.2. (a) Let sk ∈ [0, 1] be a dyadic number. Then x (γ(sk)) ∈ D(A) and

N(sk) X ∂S ϕ(τ) Axv(τ(s )) = λ Sv(s) i + S (τ) k i i ϕ(τ) i i=1 Z 1 v v = lim (x (γ(sk)) − x (τ(sk))) Pτ (Xt ∈ dγ). t→0 t 44 4 Chapter 4-Main results

(b) Let v ∈ {1, . . . , d}. Suppose (4.15) and

∞ 2 X λ p d2 < ∞. (4.23) 2p p=0

Then xv(γ(s)) ∈ D(A) and ∞ X ∂S ϕ(τ) Axv(τ(s )) = λ Sv(s) i + S (τ) k i i ϕ(τ) i i=1 Z 1 v v = lim (x (γ(sk)) − x (τ(sk))) Pτ (Xt ∈ dγ). t→0 t Proof: (a) We note

1 1 Z Z ∞ v v X v 2 x (γ(sk)) = χ[0,sk](u)dγu = hχ[0,sk],HiiL Hi(u)dγu 0 0 i=1 1 ∞ Z ∞ X v X 2 = hχ[0,sk],HiiL Hi(u)dγu = h(sk ∧ ·)ev,SiiHSi(γ). i=1 0 i=1

Since sk is dyadic the sum is ﬁnite and we get

N(sk) N(sk) v X X v x (γ(sk)) = h(sk ∧ ·)ev,SiiHSi(γ) = Si (sk)Si(γ). i=1 i=1 (a) follows immediately. v P∞ v P∞ v (b) As in (a) we have x (γ(s)) = i=1 Si (s)Si(γ). Therefore i=1 λiSi (s)Si con- 2 2 P∞ 2 v 2 verges in L (ν) and therefore also in L (ϕν) if i=1 λi Si (s) < ∞. We also have that P∞ v 2 P∞ v i=1 λiSi (s)∂Si ϕ/ϕ converges in L (ϕν) if i=1 λiSi (s) < ∞. Using Lemma 4.5.5 and (4.23) the result follows. Remark 4.8.3. For f, g ∈ Y , or f, g ∈ Z if (4.15) holds, we have for g ∈ D(E) ∩ L∞, where D(E) denotes DY (E) or DZ (E) respectively Z gΓ(f, f) ϕdν = −E(g, f 2) + 2E(fg, f) Z ∞ X 2 = − λi(∂Si f )(∂Si g) ϕdν i=1 ∞ Z X + 2 λi(∂Si (fg))(∂Si f) ϕdν i=1 Z ∞ ! X 2 = g 2 λi(∂Si f) ϕdν i=1 4.8 Local ﬁrst moment 45 where Γ denotes the carré du champ operator (see [1] p. 17). Thus for f ∈ Y ∩ L∞, or f ∈ Z ∩ L∞ if (4.15) holds, we have

∞ X 2 Γ(f, f) = 2 λi(∂Si f) . i=1 Using Proposition 4.8.1 we obtain Z 1 2 lim kSi(γ) − Si(τ)k d Pτ (Xt ∈ dγ) = 2λi t→0 t R which formally coincides with Proposition 5.1 in [4]. The same holds for linear combinations of Si(γ) belonging to D(E).

5 Chapter 5-Notation and basic deﬁnitions

5.1 Introduction

This chapter contains a selection of deﬁnitions and theorems that the reader is assumed to be familiar with from earlier experience. The reader could also read this section to refresh their memory. The section also has a list of notation used throughout the thesis.

5.2 Deﬁnitions

Deﬁnition 5.2.1. Given a measurable space (X, Σ) and a measure µ on that space, a set A is called an atom if µ(A) > 0 and for any measurable set B ⊂ A

µ(B) = 0.

If the measure µ has no atoms it is called non-atomic or atomless.

Deﬁnition 5.2.2. A Gaussian random variable X is centered if

E[X] = 0.

Deﬁnition 5.2.3. A topological space X is called separable if it contains a countable ∞ dense subset Y . In other words there exists a sequence {yn}n=1 in Y such that every nonempty open subset of X contains at least one element of this sequence.

Deﬁnition 5.2.4. A probability space (Ω, F,P ) is complete if for all B ∈ F such that P (B) = 0 one has that for every A ⊂ B that A ∈ F.

47 48 5 Chapter 5-Notation and basic deﬁnitions

Deﬁnition 5.2.5. Let X,Y be Banach spaces. A linear operator A : D ⊂ X → Y is called closed if for every sequence (xn) ∈ D(A) s.t. xn → x ∈ X and Axn → y ∈ Y one has x ∈ D(A) and Ax = y. Deﬁnition 5.2.6. Let X,Y be Banach spaces. A linear operator A : D ⊂ X → Y is called closable if for every sequence (xn) ∈ D(A) s.t. xn → 0 and Axn → y, one has y = 0.

Deﬁnition 5.2.7. A set of functions {fα : α ∈ A} : X → R is called a separating set for X or said to separate the points of X if for any two distinct elements x and y in X there exists a function f ∈ {fα : α ∈ A} such that f(x) 6= f(y).

Theorem 5.2.8. (Banach-Saks theorem) Let H be a Hilbert space and xn ∈ H, n ∈ N be bounded sequence, i.e. kxnk ≤ C for all n ∈ N. Then there exists a subsequence and an element x ∈ H such that

N 1 X x → x, as N → ∞. N nk k=1

5.3 Notation

The same symbol can be used for different purposes. The principal notation is listed below, any deviations from this is explained in the text.

Symbols and Operators

h·, ·iH The inner product in H k · kH The norm in H · ∧ ◦ Minimum of ·, ◦ A The generator 1A The characteristic function of the set A B The Borel σ-algebra ∞ Cp Smooth functions with polynomial growth D The Malliavin gradient operator Dh Directional derivative D1,p Sobolev space (we have DF ∈ Lp) δ The divergence operator Dom The domain D(·) The domain E[·] Expected value ei Element of an ON-basis E A bilinear form En The set of elementary functions F A σ-algebra 5.3 Notation 49

Gα An element in the resolvent H A Hilbert space. Often L2([0, 1]) Hn The nth Hermite polynomial, the nth Haar function Hn The nth Wiener chaos In Stochastic integral of order n Jn The projection operator onto the nth Wiener chaos Λ The set of finite multiindices L The generator of the Ornstein-Uhlenbeck semigroup Ls Set of symmetric functions N The natural numbers {0, 1,... } d Ω A sample space or the space C0([0, 1); R ) Φ Density of a normal distribution Φa A RV defined through Hermite polynomials ρ(·) The resolvent set R The real numbers Γ The carré du champ operator Si The ith Schauder function Si Random variable linked to Si Tt An element of the semigroup µ A measure, often Lebesgue measure ν A measure, often Wiener measure V [·] The variance Wt Wiener process at time t ω An element of Ω Y A set of cylinder functions defined on dyadic points Z A set of cylinder functions

ZH Cylinder functions with values in H

Abbreviations and Acronyms

a.e. Almost everywhere a.s. Almost surely i.i.d. Independent and identically distributed ON Orthonormal RV Random variable s.t. Such that w.l.o.g. Without loss of generality w.r.t. With respect to 50 5 Chapter 5-Notation and basic deﬁnitions Bibliography

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