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On a Family of Generalized Wiener Spaces and Applications University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Papers in Mathematics Mathematics, Department of Spring 5-2011 On a Family of Generalized Wiener Spaces and Applications Ian Pierce University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathstudent Part of the Analysis Commons, and the Science and Mathematics Education Commons Pierce, Ian, "On a Family of Generalized Wiener Spaces and Applications" (2011). Dissertations, Theses, and Student Research Papers in Mathematics. 33. https://digitalcommons.unl.edu/mathstudent/33 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Dissertations, Theses, and Student Research Papers in Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. ON A FAMILY OF GENERALIZED WIENER SPACES AND APPLICATIONS by Ian D. Pierce A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professor David L. Skoug Lincoln, Nebraska May, 2011 ON A FAMILY OF GENERALIZED WIENER SPACES AND APPLICATIONS Ian D. Pierce, Ph. D. University of Nebraska, 2011 Adviser: David L. Skoug We investigate the structure and properties of a variety of generalized Wiener spaces. Our main focus is on Wiener-type measures on spaces of continuous functions; our generalizations include an extension to multiple parameters, and a method of adjusting the distribution and covariance structure of the measure on the underlying function space. In the second chapter, we consider single-parameter function spaces and extend a fun- damental integration formula of Paley, Wiener, and Zygmund for an important class of functionals on this space. In the third chapter, we discuss measures on very general function spaces and introduce the specific example of a generalized Wiener space of several parame- ters; this will be the setting for the fourth chapter, where we extend some interesting results of Cameron and Storvick. In the final chapter, we apply the work of the preceding chapters to the question of reflection principles for single-parameter and multiple-parameter Gaussian stochastic processes. iii COPYRIGHT c 2011, Ian D. Pierce iv ACKNOWLEDGMENTS I owe a great deal of thanks and acknowledgement to many people for inspiration, assistance, and patient forbearance. At the risk of omitting some, I particularly note the following: I thank my advisor, David Skoug, who has been very patient with me and endured much aggravation in shepherding me to this point. I also thank Jerry Johnson and Lance Nielsen, for their helpful comments and suggestions, for enduring my endless seminar presentations, and for going the extra ten miles in reading and commenting on this work. To the other mem- bers of my committee, I also extend my appreciation: to Sharad Seth and Allan Peterson, and also to Mohammad Rammaha, who taught me much about Analysis. I am grateful to my entire family, my friends, and my former teachers for their help and encouragement through the years; my parents certainly deserve special mention in this. Finally, I must thank my own little family - Maria, Patrick, and \Baby Joe" - for their support, and for bearing with me and tolerating my absence and lack of attention during the completion of this dissertation. Whatever is of quality in this dissertation is built upon the foundation, earnest efforts, and good offices of those who preceded and assisted me; any failures or shortcomings are mine alone. I lay this work at the feet of God, the Father, Son, and Holy Spirit, to whom be all glory and honor unto the ages of ages. Amen. v Contents Contents v 1 Background and Introduction 1 1.1 Background . 1 1.2 Overview . 5 1.3 Questions for Future Work . 8 2 Single Parameter Spaces 10 2.1 An Introduction to the Function Space Ca;b[0;T ] . 10 2.2 Stochastic Integrals on Ca;b[0;T ]......................... 19 2.3 Paley-Wiener-Zygmund Theorem . 31 3 General Spaces 41 3.1 Cylinder sets and cylindrical Gaussian measures . 41 3.2 Centered Gaussian measures on Lebesgue spaces . 46 3.3 General construction and properties . 51 3.4 Measures on C(S)................................. 64 3.5 Measurability . 77 3.6 Bounded Variation and Absolute Continuity . 78 3.7 Measures on the space C0(Q) .......................... 83 vi 3.8 Examples . 98 4 Integration Over Paths 104 4.1 Preliminaries . 106 4.2 One-line Theorems . 111 4.3 n-line Theorem . 113 4.4 Applications and Examples . 117 5 Reflection Principles 119 5.1 Introduction . 119 5.2 A reflection principle for the general function space Ca;b[0;T ] . 120 5.3 Reflection principles for two parameter Wiener space . 127 5.4 A positive reflection result for C2(Q) . 137 Bibliography 141 1 Chapter 1 Background and Introduction 1.1 Background For finite-dimensional spaces, Lebesgue measure serves as a canonical example; it has the important (and intuitive) properties of scale and translation invariance. However, it is well-known that there can be no reasonable translation invariant measure on an infinite- dimensional space. The problem of defining a somewhat reasonable measure on such a space was first solved by Norbert Wiener in [58]. Wiener constructed his measure on the space C[0; 1] of continuous functions on the unit interval. Elements of the support of this measure satisfy the conditions to be sample paths of an ordinary Brownian motion; thus Wiener's work made mathematically rigorous the model proposed by Albert Einstein (thus foreshadowing the significant applications to problems in mathematical physics which were to follow). A standard Brownian motion is a model originally intended to describe the apparently random motion of a particle moving in a liquid. For a Brownian motion, the particle trajec- tories must satisfy several conditions, the first being continuity. In addition, the position at time t must be normally distributed with mean 0 and variance t, and this distribution must 2 have independent increments. Viewing the path of a particle as a stochastic process and using these properties, one can show that the transition probabilities for a discretization of the process using the partition 0 = t0 < t1 < : : : < tn = 1 have density functions 2 1 (uj+1 − uj) p exp − ; 2π(tj+1 − tj) 2(tj+1 − tj) as per Einstein's proposed model. Wiener's task was to start with sets defined in terms of finite discretizations of this type, and from them to obtain a countably additive set function on the Borel algebra of subsets of C[0;T ]. This is quite involved, and is all the more impressive when one recalls that the Lebesgue Theory was really quite new at the time, and that much of the modern machinery of functional analysis was unavailable to Wiener. From the basic facts discussed above we can easily obtain the ability to integrate \tame" n functionals of the form F (x) = f(x(t1); : : : ; x(tn)), where f : R ! C is Lebesque measurable against Wiener's measure w using the formula 1 n !− Z Y 2 Z F (x)w(dx) = (2π(tj − tj−1)) f(u1; : : : ; un) n C[0;1] j=1 R n 2 ! 1 X (uj − uj−1) exp − du ··· du : (1.1) 2 t − t n 1 j=1 j j−1 A considerable amount of work was accomplished with little more than this formula and very clever estimation and limiting arguments, and the additional theorem from [42] of Paley, Wiener, and Zygmund stating that for fh1; : : : ; hng an orthonormal collection (in the L2-sense) of functions of bounded variation on [0; 1] and for F : C[0; 1] ! C defined by 3 R 1 R 1 F (x) = f 0 h1dx; : : : ; 0 hndx , the following holds: Z Z n 2 ! − n X uj F (x)w(dx) = (2π) 2 f(u1; : : : ; un) exp − dun ··· du1: (1.2) n 2 C[0;1] R j=1 Perhaps the best-known contributers to the field following Paley, Wiener, and Zygmund were R.H. Cameron and W.T. Martin and their disciples. Of great significance was the Cameron-Martin translation theorem, which drew on the underlying structure hinted at by (1.2). This translation theorem showed that while the Wiener measure was not translation invariant, it was quasi-invariant with respect to translation by a certain subset of C[0; 1]. Now called the Cameron-Martin space, this is the collection of absolutely continuous functions 2 2 with L derivatives, ( i.e. the Sobolev space H0 [0; 1]). If T : C[0; 1] ! C[0; 1] is defined by 0 x 7! x + x0, where x0 is in the Cameron-Martin Space and x0 is of bounded variation, then the translation theorem asserts that the Radon-Nicodym derivative of the measure γ ◦ T with respect to the untranslated measure γ is given by Z 1 d(γ ◦ T ) 1 0 2 0 (x) = exp − jjx0jj2 − x0(t)dx(t) : (1.3) dγ 2 0 0 The condition that x0 be of bounded variation can be relaxed by substituting a stochastic integral for the Riemann-Stieltjes integral in (1.3). This translation theorem has many significant implications. As a connection to the theory of probability, we note also that this is a special case of the Girsanov Theorem. Considerable development and generalization was accomplished by many of Cameron's students and associates; of particular note for my research have been Donsker, Baxter, Varberg, Kuelbs, Yeh, C. Park, and Skoug. Two forms of generalization are of particular interest to us; the first is in considering a wider variety of transition probability densities and the second is in allowing for parameter sets other than the interval [0; 1].
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