This book offers a concise introduction to the rapidly expanding field of infinite dimensional stochastic analysis.It treats Malliavin calculus and whitenoise analysis in a single book,presenting these two different areas in a unified setting of Gaussian Probability spaces.Topics include tecent results and developments in the areas of quasis-sure analysis,anticipating stochastic calculus,generalised operator theory and applications in quantum physics ,A short overview on the foundations of infinite dimensional analysis is given. Audience This volume will be of interest to researchers and graduate students whose work involves probability rheory,stochastic processes,functional analysis,operator theory,mathematics of physics and abstract harmonic analysis.
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目录
Preface Chapter I Foundations of Infinite Dimensional Analysis 1 Linear operators on Hilbert spaces 11 Basic notions, notations and lemmas 12 Closable, symmetric and self-adjoint operators 13 Self-adjoint extension of a symmetric bounded below operator 14 Spectral resolution of self-adjoint operators 15 Hilbert-Schmidt and trace class operators 2 Fock spaces and second quantization 21 Tensor products of Hilbert spaces 22 Fock spaces 23 Second quantization of operators 3 Countably normed spaces and nuclear spaces 31 Countably normed spaces and their dual spaces 32 Nuclear spaces and their dual spaces 33 Topological tensor product, the Schwartz kernels theorem 4 Borel measures on topological linear spaces 41 Minlos-Sazanov theorem 42 Gaussian measures on Hilbert spaces 43 Gaussian measures on Banach spaces Chapter II Malliavin Calculus 1 Gaussian probability spaces and \\Viener chaos decomposition 11 Functionals on Gaussian probability spaces 12 Numerical models 13 Multiple Wiener-lto integral representation 2 Differential calculus of functionals, gradient and divergence operators 21 Finite dimensional Gaussian probability spaces 22 Gradient and divergence of smooth functionals 23 Sobolev spaces of functionals 3 Meyers inequalities and some consequences 31 Ornstein-Uhlenbeck semigroup 32 i-multiplier theorem 33 Meyers inequalities 34 Meyer-Watanabes generalized functionals 4 Densities of non-degenerate functionals 41 Malliavin covariance matrices, some lemmas 42 Existence of densities 43 Smoothness of densities 44 Examples Chapter III Stochastic Calculus of Variation for \\Viener Functionals 1 Differential calculus of lto functionals and regularity of heat kernels 11 Skorohod integrals 12 Smoothness of solutions to stochastic differential equations 13 Hypoellipticity and Hormanders conditions 14 A probabilistic proof of Hormanders theorem 2 Potential theory over Wiener spaces and quasi-sure analysis 21 (fc,p)-capacities 22 Quasi-continuous modifications 23 Tightness, continuity and invariance of capacities 24 Positive generalized functionals and measures with finite energy 25 Some quasi-sure sample properties of stochastic processes 3 Anticipating stochastic calculus 31 Approximation of Skorohod integrals by Riemannian sums 32 lto formula for anticipating processes 33 Anticipating stochastic differential equations Chapter IV General Theory of White Noise Analysis 1 General framework for white noise analysis 11 Wick tensor products and the Wiener-lto-Segal isomorphisn 12 Testing functional space and distribution space 13 Classical framework for white noise analysis 2 Characterization of functional spaces 21 -transform and characterization of space {E){o<,/3 22 Local -transform and characterization of space {E) 23 Two characterizations for testing functional spaces 24 Some examples of distributions Contents 3 Products and AVick products of functionals 31 Products of functionals 32 Wick products of distributions 33 Application to Feynman integrals 4 Moment characterization of distributions and positive distributions 41 The renormalization operator 42 Moment characterization of distribution spaces 43 Measure representation of positive distributions 44 Application to P()2-quantum fields Chapter V Linear Operators on Distribution Spaces 1 Analytic calculus for distributions 11 Scaling transformations 12 Shift operators and Sobolev differentiations 13 Gradient and divergence operators 2 Continuous linear operators on distribution spaces 21 Symbols and chaos decompositions for operators 22 5-transforms and Wick products of generalized operators 3 Integral kernel operators and integral kernel representation for operators 31 Contraction of tenser products 32 Integral kernel operators 33 Integral kernel representation for generalized operators 4 Applications to quantum physics 41 Quantum stochastic integrals 42 Klein-Gordon field 43 Infinite dimensional classical Dirichlet forms Appendix A Hermite polynomials and Hermite functions Appendix B Locally convex spaces and their dual spaces 1 Semi-norms, norms and H-norms 2 Locally convex topological linear spaces, bounded sets 3 Projective topologies and projective limits 4 Inductive topologies and inductive limits 5 Dual spaces and weak topologies 6 Compatibility and Mackey topology 7 Strong topologies and reflexivity 8 Dual maps 9 Uniformly convex spaces and Banach-Saks theorem Comments References Subject Index Index of Symbols
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