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Lecture Notes in Mathematics 1610 Editors: A Lecture Notes in Mathematics 1610 Editors: A. Dold, Heidelberg F. Takens, Groningen Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo Ali Stileyman Usttinel An Introduction to Analysis on Wiener Space Springer Author Ali Stileyman lJsttinel ENST, Drpt. Rrseaux 46, rue Barrault F-75013 Paris, France e-mail: [email protected] Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Usflinel, All Siileyman: An introduction to analysis on Wiener space / All SOleyman Ustt~nel, - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 1995 (Lecture notes in mathematics ; Vol 1610) ISBN 3-540-60170-8 NE: GT Mathematics Subject Classification (1991): 60H07, 60H05, 60H15, 46F25, 81T, 81Q ISBN 3-540-60170-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10479544 46/3142-543210 - Printed on acid-free paper Ankara Fen Lisesi 67-70 mezunlarma. Introduction The following pages are the notes from a seminar that I gave during the spring and some portion of the summer of 1993 at the Mathematics Institute of Oslo University. The aim of the seminars was to give a rapid but rigorous introduction for the graduate students to Analysis on Wiener space, a subject which has grown up very quickly these recent years under the impulse of the Stochastic Calculus of Variations of Paul Malliavin (cf. [12]). Although some concepts are in the preliminaries, I assumed that the students had already acquired the notions of stochastic calculus with semimartingales, Brownian motion and some rudiments of the theory of Markov processes. A small portion of the material exposed is our own research, in particular, with Moshe Zakai. The rest has been taken from the works listed in the bibliography. The first chapter deals with the definition of the (so-called) Gross-Sobolev derivative and the Ornstein-Uhlenbeck operator which are indispensable tools of the analysis on Wiener space. In the second chapter we begin the proof of the Meyer inequalities, for which the hypercontractivity property of the Ornstein- Uhlenbeck semigroup is needed. We expose this last topic in the third chapter, then come back to Meyer inequalities, and complete their proof in chapter IV. Different applications are given in next two chapters. In the seventh chapter we study the independence of some Wiener functionals with the previously devel- oped tools. The chapter VIII is devoted to some series of moment inequalities which are important for applications like large deviations, stochastic differential equations, etc. In the last chapter we expose the contractive version of Ramer's theorem as another example of the applications of moment inequalities developed in the preceding chapter. During my visit to Oslo, I had the chance of having an ideal environment for working and a very attentive audience in the seminars. These notes have partic- ularly profited from the serious criticism of my colleagues and friends Bernt Oksendal, Tom LindstrOm, Ya-Zhong Hu, and the graduate students of the Mathematics department. It remains for me to express my gratitude also to Nina Haraldsson for her careful typing, and, last but not least, to Laurent De- creusefond for correcting so many errors . Ali Sfileyman Ostiinel Contents Preliminaries 1 1 The Brownian Motion and the Wiener Measure .......... 1 Gross-Sobolev Derivative, Divergence and Ornstein-Uhlenbeck Operator 9 1 The Construction of V and its properties ............. 10 1.1 Relations with the stochastic integration .......... 12 2 The Ornstein-Uhlenbeck Operator ................. 16 II Meyer Inequalities 19 1 Some Preparations .......................... 19 2 ~(I + f~)-1/2 as the Riesz Transform ............... 21 III Hypercontractivity 27 IV LP-Multipliers Theorem, Meyer Inequalities and Distributions 31 1 LP-Multipliers Theorem ....................... 31 V Some applications of the distributions 41 1 Extension of the Ito-Clark formula ................. 42 2 Lifting of,_q'(~d) with random variables .............. 45 VI Positive distributions and applications 53 1 Capacities and positive Wiener functionals ............. 55 VII Characterization of independence of some Wiener functionals 61 1 Independence of Wiener functionals ................ 61 VIII Moment inequalities for Wiener functionals 69 1 Exponential tightness ........................ 60 2 Coupling inequalities ......................... 73 3 An interpolation inequality ..................... 77 IX Introduction to the theorem of Ramer 81 Bibliography 91 Subject Index 94 Notations 95 Preliminaries This chapter is devoted to the basic results about the Wiener measure, Brownian motion, construction of the Ito stochastic integral and the chaos decomposition associated to it. 1 The Brownian Motion and the Wiener Mea- sure 1) Let W = C0([O, 1]), w E W, t E [0, 1], define Wt(w) = w(t) (the coordinate functional). If we note by Bt = a{Ws;s < t}, then there is one and only one measure tt on W such that i) tt{W0(w) = O} = 1, ii) Vf E C~(R), the stochastic process process (t,~) ~ S(w,(~)) - ~1/0' S"(W,(~))ds is a (Bt, ~)-martingale. /~ is called the Wiener measure. 2) From the construction we see that for t > s, E•[exp ia(Wt - W,)IB, ] = exp-c~2(t - s), hence (t, w) ~ Wt(w) is a continuous additive process (i.e.,a process with inde- pendent increments) and (Wt;t E [0, 1]) is also a continous martingale. 3) Stochastic Integration Let K : W x [0, 1] --* R be a step process : n Kt(w) = Eai(w). l[t,.t,+,t(t), ai(w) E L2(Bt,) 9 i=1 Define I(K) = fO K~dW,(w) 2 Preliminaries as n (W,,+, - i----1 Then we have i.e. I is an isometry from the adapted step processes into L~(p), hence it has a unique extension as an isometry from L2([0,1] x W,A, dt x dp) ;~ L~(p) where .A denotes the sigma algebra on [0, 1] x W generated by the adapted, left (or right) continuous processes. I(K) is called the stochastic integral of K and it is denoted as .[1 KjdWs. If we define It(K) = ~0 t K, dWs as 1 fo l[o,tl(s)K, dWs, it is easy to see that the stochastic process t ~ It(K) is a continuous, square integrable martingale. With some localization techniques using stopping times, I can be extended to any adapted process K such that f: K~(w)ds < c~ a.s. In this case the process t ~ It(K) becomes a local martingale, i.e., there exists a sequence of stopping times increasing to one, say (Tn, n E N) such that the process t ~ hAT. (K) is a (square integrable) martingale. Application: Ito formula We have following important applications of the stochastic integration: a) If f E C2(R) and Mt = f~ grdWr, then f(Mt) = f(O) + f[ f'(Ms)KsdWs + -~1/' f"(M,)K~ds. b) gt(I(h)) = exp( ~ot h, dW, - -~1 ~0t h~ds) is a martingale for any h E L 2 [0, 1]. Prelimin aries 3 4) Alternative constructions of the Brownian motion and the Wiener measure A) Let (Ti;i E N) be an independent sequence of Nt(0, 1) Gaussian random variables. Let (gi) be a complete, orthonormal basis of L2[0, 1]. Then Wt defined by Wt(w) = ETi(w) 9 gi(s)ds i=l fo' is a Brownian motion. Remark: If (gi;i E N) is a complete, orthonormal basis of L2([0, 1]), then (fogi(s)ds;i E N) is a complete orthonormal basis of H([0, 1])(i. e., the first order Sobolev functionals on [0, 1]). B) Let (f~,.T, P) be any abstract probability space and let H be any separable Hilbert space. If L : H ~ L2(~,~', P) is a linear operator such that for any h E H, E[exp iL(h)l = exp-89[hl~r, then there exists a Banach space with dense injection H ~ W dense, hence W* ~ H is also dense and a probability measure /~ on W such that exp(w*,w)dp(w) = exp-89I J*@*) and L(j*(w*))(w) = (w*,w) almost surely. (IV, H, p) is called an Abstract Wiener space and p is the Wiener measure. If H([0, 1]) = {h : h(t) = ft h(s)ds, [hill = ]hiL~[0,1]} then p is the classical Wiener measure and W can be taken as C0([0, 1]). Remark: In the case of the classical Wiener space, any element X of W* is a signed measure on [0, 1], and its image in H -- H([0, 1]) can be represented as j*(X)(/) = fj X([s, 1])ds. In fact, we have for any h E H (j*(X),h) = <A,j(h)> = h(s)a(es) = h(1)a([O, 1])- a([O, sl)h(s)ds = /o 1(A([O, 1]) - A([O, s])88 -- /o A([s, 1])h(s)ds. 4 Preliminaries 5) Let us come back to the classical Wiener space: i) It follows from the martingale convergence theorem and the monotone class theorem that the set of random variables {f(W~l,... , Wt.);ti E [0, 1], f E S(Rn);n E N} is dense in L 2(/z), where S(R n) denotes the space of infinitely differentiable, rapidly decreasing functions on R n.
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