Introduction to Infinite Dimensional Stochastic Analysis

Transcription

1 Introduction to Infinite Dimensional Stochastic Analysis By Zhi yuan Huang Department of Mathematics, Huazhong University of Science and Technology, Wuhan P. R. China and Jia an Yan Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing P. R. China. Illl Science Press Beijing/New York, tf KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON/LONDON

2 Contents Preface ix Chapter I Foundations of Infinite Dimensional Analysis 1 1. 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 51 Chapter II Malliavin Calculus Gaussian probability spaces and Wiener chaos decomposition Functional 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 Omstein-Uhlenbeck semigroup 86

3 VI Contents 3.2 z, p -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 110 Chapter III Stochastic Calculus of Variation for Wiener Functionals Differential calculus of ltd 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 (fc,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 155 Chapter IV General Theory of White Noise Analysis General framework for white noise analysis 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 (B)~ /3 (o</3<i) Local s-transform and characterization of space (E)^ Two characterizations for testing functional spaces Some examples of distributions 183

4 Contents 3. Products and Wick products of functionals Products of functionals Wick productsof 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(0) 2 -quantum fields 206 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 245 vii Appendix A Hermite polynomials and Hermite functions 252 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 263

5 Vlll Contents 8. Dual maps Uniformly convex spaces and Banach-Saks' theorem 264 Comments 266 References 271 Subject Index 290 Index of Symbols 294