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
Contents Preface ix Chapter I Foundations of Infinite Dimensional Analysis 1 1. Linear operators on Hilbert spaces 1 1.1 Basic notions, notations and lemmas 1 1.2 Closable, symmetric and self-adjoint operators 4 1.3 Self-adjoint extension of a symmetric bounded below operator... 8 1.4 Spectral resolution of self-adjoint operators 10 1.5 Hilbert-Schmidt and trace class operators 14 2. Fock spaces and second quantization 19 2.1 Tensor products of Hilbert spaces 19 2.2 Fock spaces 24 2.3 Second quantization of operators 26 3. Countably normed spaces and nuclear spaces 29 3.1 Countably normed spaces and their dual spaces 30 3.2 Nuclear spaces and their dual spaces 34 3.3 Topological tensor product, the Schwartz kernels theorem 38 4. Borel measures on topological linear spaces 41 4.1 Minlos-Sazanov theorem 41 4.2 Gaussian measures on Hilbert spaces 48 4.3 Gaussian measures on Banach spaces 51 Chapter II Malliavin Calculus 59 1. Gaussian probability spaces and Wiener chaos decomposition 59 1.1 Functional on Gaussian probability spaces 59 1.2 Numerical models 64 1.3 Multiple Wiener-Ito integral representation 67 2. Differential calculus of functionals, gradient and divergence operators 72 2.1 Finite dimensional Gaussian probability spaces 72 2.2 Gradient and divergence of smooth functionals 76 2.3 Sobolev spaces of functionals 81 3. Meyer's inequalities and some consequences 86 3.1 Omstein-Uhlenbeck semigroup 86
VI Contents 3.2 z, p -multiplier theorem 89 3.3 Meyer's inequalities 92 3.4~Meyer-Watanabe's generalized functionals 97 4. Densities of non-degenerate functionals 100 4.1 Malliavin covariance matrices, some lemmas 101 4.2 Existence of densities : 103 4.3 Smoothness of densities 106 4.4 Examples 110 Chapter III Stochastic Calculus of Variation for Wiener Functionals 113 1. Differential calculus of ltd functionals and regularity of heat kernels 113 1.1 Skorohod integrals 113 1.2 Smoothness of solutions to stochastic differential equations 118 1.3 Hypoellipticity and Hormander's conditions 120 1.4 A probabilistic proof of Hormander's theorem 125 2. Potential theory over Wiener spaces and quasi-sure analysis 130 2.1 (fc,p)-capacities 130 2.2 Quasi-continuous modifications 133 2.3 Tightness, continuity and invariance of capacities 135 2.4 Positive generalized functionals and measures with finite energy 139 2.5 Some quasi-sure sample properties of stochastic processes 142 3. Anticipating stochastic calculus 145 3.1 Approximation of Skorohod integrals by Riemannian sums 145 3.2 Ito formula for anticipating processes 149 3.3 Anticipating stochastic differential equations 155 Chapter IV General Theory of White Noise Analysis 161 1. General framework for white noise analysis 162 1.1 Wick tensor products and the Wiener-Ito-Segal isomorphism.. 162 1.2 Testing functional space and distribution space... 165 1.3 Classical framework for white noise analysis 169 2. Characterization of functional spaces 171 2.1 s-transform and characterization of space (B)~ /3 (o</3<i) 171 2.2 Local s-transform and characterization of space (E)^. 1 177 2.3 Two characterizations for testing functional spaces 179 2.4 Some examples of distributions 183
Contents 3. Products and Wick products of functionals 188 3.1 Products of functionals 188 3.2 Wick productsof distributions 191 3.3 Application to Feynman integrals 193 4. Moment characterization of distributions and positive distributions 195 4.1 The renormalization operator 195 4.2 Moment characterization of distribution spaces 197 4.3 Measure representation of positive distributions 199 4.4 Application to p(0) 2 -quantum fields 206 Chapter V Linear Operators on Distribution Spaces 210 1. Analytic calculus for distributions 210 1.1 Scaling transformations 210 1.2 Shift operators and Sobolev differentiations 212 1.3 Gradient and divergence operators 216 2. Continuous linear operators on distribution spaces 219 2.1 Symbols and chaos decompositions for operators 219 2.2 s-transforms and Wick products of generalized operators 224 3. Integral kernel operators and integral kernel representation for operators 229 3.1 Contraction of tensor products 229 3.2 Integral kernel operators 231 3.3 Integral kernel representation for generalized operators 237 4. Applications to quantum physics 240 4.1 Quantum stochastic integrals 240 4.2 Klein-Gordon field 243 4.3 Infinite dimensional classical Dirichlet forms 245 vii Appendix A Hermite polynomials and Hermite functions 252 Appendix B Locally convex spaces and their dual spaces 257 1. Semi-norms, norms and H-norms 257 2. Locally convex topological linear spaces, bounded sets 258 3. Projective topologies and projective limits 259 4. Inductive topologies and inductive limits 260 5. Dual spaces and weak topologies 261 6. Compatibility and Mackey topology 262 7. Strong topologies and reflexivity 263
Vlll Contents 8. Dual maps 263 9. Uniformly convex spaces and Banach-Saks' theorem 264 Comments 266 References 271 Subject Index 290 Index of Symbols 294