Introduction to Infinite Dimensional Stochastic Analysis

Transcription

1 Introduction to Infinite Dimensional Stochastic Analysis

2 Mathematics and Its Applications Managing Editor M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 502

3 Introduction to Infinite Dimensional Stochastic Analysis By Zhi-yuan Huang Department of Mathematics, Huazhong University ofscience and Technology, Wuhan P. R. China and Jia-an Yan Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing P. R. China. Science Press Beijing/New York, SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

4 A C.I.P Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / This is an updated and revised translation of the original Chinese publicat ion Science Press, Beijing, P. R. China, Printed an acid-free paper AII Rights Reserved 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 Softcover reprint ofthe hardcover lst edition 2000 N o part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any informat ion storage and retrieval system, without written permission from the copyright owners.

5 Contents Preface... 1X Chapter I Foundations of Infinite Dimensional Analysis Linear operators on Hilbert spaces Basic notions, notations and lemmas Closable, symmetric and self-adjoint operators Self-adjoint extension of a symmetric bounded below operator Spectral resolution of self-adjoint operators Hilbert-Schmidt and trace class operators Fock spaces and second quantization Tensor products of Hilbert spaces Fock spaces Second quantization of operators Countably normed spaces and nuclear spaces Countably normed spaces and their dual spaces Nuclear spaces and their dual spaces Topological tensor product, the Schwartz kernels theorem Borel measures on topological linear spaces Minlos-Sazanov theorem Gaussian measures on Hilbert spaces Gaussian measures on Banach spaces Chapter II Malliavin Calculus Gaussian probability spaces and Wiener chaos decomposition Functionals on Gaussian probability spaces Numerical models Multiple Wiener-Ito integral representation Differential calculus of functionals, gradient and divergence operators Finite dimensional Gaussian probability spaces Gradient and divergence of smooth functionals Sobolev spaces of functionals Meyer's inequalities and some consequences Ornstein-Uhlenbeck semigroup... 86

6 VI Contents 3.2 LP-multiplier theorem Meyer's inequalities Meyer-Watanabe's generalized functionals Densities of non-degenerate functionals Malliavin covariance matrices, some lemmas Existence of densities Smoothness of densities Examples Chapter III Stochastic Calculus of Variation for Wiener Functionals Differential calculus of Ito functionals and regularity of heat kernels Skorohod integrals Smoothness of solutions to stochastic differential equations Hypoellipticity and Hormander's conditions A probabilistic proof of Hormander's theorem Potential theory over Wiener spaces and quasi-sure analysis (k,p)-capacities Quasi-continuous modifications Tightness, continuity and invariance of capacities Positive generalized functionals and measures with finite energy Some quasi-sure sample properties of stochastic processes Anticipating stochastic calculus Approximation of Skorohod integrals by Riemannian sums Ito formula for anticipating processes Anticipating stochastic differential equations Chapter IV General Theory of White Noise Analysis General framework for white noise analysis ;1 Wick tensor products and the Wiener-Ito-Segal isomorphism Testing functional space and distribution space Classical framework for white noise analysis Characterization of functional spaces s-transform and characterization of space (E)~/(O::;,8<I) Local s-transform and characterization of space (Ere Two characterizations for testing functional spaces Some examples of distributions

7 Contents vii 3. Products and Wick products of functionals Products of functionals Wick products of distributions Application to Feynman integrals Moment characterization of distributions and positive distributions The renormalization operator Moment characterization of distribution spaces Measure representation of positive distributions Application to p(4)).-quantum fields Chapter V Linear Operators on Distribution Spaces Analytic calculus for distributions Scaling transformations Shift operators and Sobolev differentiations Gradient and divergence operators Continuous linear operators on distribution spaces Symbols and chaos decompositions for operators s-transforms and Wick products of generalized operators Integral kernel operators and integral kernel representation for operators Contraction of tensor products Integral kernel operators Integral kernel representation for generalized operators Applications to quantum physics Quantum stochastic integrals Klein-Gordon field Infinite dimensional classical Dirichlet forms Appendix A Hermite polynomials and Hermite functions Appendix B Locally convex spaces and their dual spaces Semi-norms, norms and H-norms Locally convex topological linear spaces, bounded sets Projective topologies and projective limits Inductive topologies and inductive limits Dual spaces and weak topologies Compatibility and Mackey topology Strong topologies and reflexivity

8 VIll Contents 8. Dual maps Uniformly convex spaces and Banach-Saks' theorem Comments References Subject Index Index of Symbols

9 Preface The infinite dimensional analysis as a branch of mathematical sciences was formed in the late 19th and early 20th centuries. Motivated by problems in mathematical physics, the first steps in this field were taken by V. Volterra, R. GateallX, P. Levy and M. Frechet, among others (see the preface to Levy[2]). Nevertheless, the most fruitful direction in this field is the infinite dimensional integration theory initiated by N. Wiener and A. N. Kolmogorov which is closely related to the developments of the theory of stochastic processes. It was Wiener who constructed for the first time in 1923 a probability measure on the space of all continuous functions (i.e. the Wiener measure) which provided an ideal mathematical model for Brownian motion. Then some important properties of Wiener integrals, especially the quasi-invariance of Gaussian measures, were discovered by R. Cameron and W. Martin[l, 2, 3]. In 1931, Kolmogorov[l] deduced a second order partial differential equation for transition probabilities of Markov processes with continuous trajectories (i.e. diffusion processes) and thus revealed the deep connection between theories of differential equations and stochastic processes. The stochastic analysis created by K. Ito (also independently by Gihman [1]) in the forties is essentially an infinitesimal analysis for trajectories of stochastic processes. By virtue of Ito's stochastic differential equations one can construct diffusion processes via direct probabilistic methods and treat them as functionals of Brownian paths (i.e. the Wiener functionals). This affords a possibility of using probabilistic methods to investigate deterministic differential equations and many other pure analytical problems. On the other hand, during the same decade of this century, the famous work on functional integration approach to mathematical physics derived by R. Feynman and M. Kac as well as rapid developments in quantum field theory gave new impulsions to the analysis in infinite dimension. The classical notions of functions and derivatives in finite dimensional analysis were long felt to be restrictive in mathematical physics. Notions of generalized functions and derivatives were first introduced by S. L. Sobolev[l] in 1936 to meet the needs of solving equations in mathematical physics. The theory of Sobolev spaces has played an important role in modern treatment of partial differential operators. L. Schwartz systematically developed this idea and established theory

10 x Preface of distributions. Many singular objects in classical physics such as Dirac delta functions thus obtained mathematically rigorous meanings. However, up to now there still exist a lot of intuitive notions and heuristic calculations in physics which remain meaningless from the mathematical viewpoint. To put them on a sound mathematical foundation is quite important for the development of theoretical physics and is a real challenge to mathematicians and physicists. The same situation also occurred in infinite dimensional analysis. Since many important functionals (e.g. diffusion processes regarded as Wiener functionals) are not differentiable in Fh~chet sense, it is essential to generalize the notions of functionals and differentiation in infinite dimensional spaces. In 1976, P. Malliavin[l] successfully extended the gradient, divergence and Ornstein-Uhlenbeck operators to infinite dimensional cases and created the stochastic calculus of variation (known as Malliavin calculus). Under his sense of differentiation, many important Wiener functionals become smooth (infinitely differentiable). Along this line S. Watanabe[lJ, I. Shigekawa[lJ, D. W. Stroock[lJ, P. A. Meyer[l] et al. established a Sobolev theory over infinite dimensional spaces. With fruitful applications to partial differential operators and heat kernels, to stochastic oscillatory integrals, to filtering and control of stochastic systems, etc., the Malliavin calculus has become one of the most significant successes in the field of stochastic analysis. In 1975, T. Hida launched out the white noise analysis. Since Gaussian white noise is the derivative of Brownian motion in the distribution sense, its sample space lies in the space of Schwartz distributions. By regarding Wiener functionals as functionals of white noise, Hida established a Schwartz theory over infinite dimensional spaces. With profound background in physics and successful applications to Feynman integrals as well as quantum field theory, white noise analysis has attracted more and more attention from theoretical physicists. These two frameworks of infinite dimensional analysis are essentially based on the quasi-invariance of Gaussian measures and could be unified into one setting of the so-called Gaussian probability spaces. Its origin goes back to the works on abstract integration on Hilbert spaces by I. E. Segal[l, 2] and on rigged Hilbert spaces by I. M. Gel'fand in the fifties (see Gel'fand & Vilenkin[l]), and the work on abstract Wiener spaces by L. Gross[l] in the sixties. The choice of frameworks depends naturally on the practical problems to be solved. In Malliavin calculus, for example, one requires the space of testing functionals to be rich enough so that many important functionals become smooth, while in Hida calculus, one hopes that the space of distributions would be sufficiently large to contain many singular objects in physics which are not rigorously defined so far. The relationship between these two kinds of calculus is quite similar to that of Sobolev and Schwartz theories in finite dimensions. This book is intended to offer a quick introduction to the above mentioned rapidly developing research area --infinite dimensional stochastic analysis. We

11 Preface Xl have attempted to make the book concise and self-contained so that readers, who have acquired the basic knowledge of stochastic analysis, can easily master the very core of the theory and methods and go quickly to the front of this research area. The book is divided into five chapters. The first chapter contains the basic knowledge of infinite dimensional analysis including linear operators on Hilbert spaces, Fock spaces, count ably normed spaces, nuclear spaces and their dual spaces, Borel measures on topological linear spaces which are the base of other chapters and also of independent interests. In order to make the book selfcontained and for reader's convenience, we summarize some basic notions and general results of locally convex topological linear spaces in Appendix B. The second chapter is devoted to the fundamental theory of Malliavin calculus, including the chaos decomposition and differentiation for functionals on Gaussian probability spaces, the Ornstein-Uhlenbeck semi-group, Meyer's inequalities, Sobolev spaces, existence and smoothness of densities of non-degenerate functionals and so on. Chapter III treats some important applications of Malliavin calculus. The regularity of densities of solutions to Ito's stochastic differential equations, namely the fundamental solutions to the corresponding second order parabolic partial differential equations, is derived in detail and a probabilistic proof of the celebrated Hormander theorem on hypoellipticity of differential operators is presented. The potential theory, quasi-sure analysis on Wiener spaces and the anticipating stochastic calculus are also briefly touched. The general theory of white noise analysis, which is the main goal of Chapter IV, consists of construction of general settings, characterization of functional spaces, the products and Wick products of functionals and the moment characterization of distributions. Applications to Feynman integrals, P( 1> )z-quantum fields and local times of selfintersection for Brownian motion are also briefly discussed. The last chapter is a presentation of the theory of operators on spaces of distributions (including some analytical operations on distributions) and its applications in quantum physics. Applications to infinite dimensional harmonic analysis are not touched here for the sake of length, for which we refer the readers to N. Obata[2]. 1, 4 of Chapter I, Chapters IV and V were written by J. A. Yan; 2, 3 of Chapter I, Chapters II and III and the appendices were written by Z. Y. Huang. Both authors wish to thank Prof. J. G. Ren and Dr. S. L. Luo, who provided considerable assistance with the preparation of 2 of Chapter III and 4 of Chapter V respectively and made many useful comments. This work was financially supported by the National Natural Science Foundation of China (grant no and ) and the Science Press Foundation of the Chinese Academy of Sciences to whom the authors would like to acknowledge. Z. Y. Huang and J. A. Yan, 1999