Introduction to Infinite Dimensional Stochastic Analysis

• I Foundations of Infinite Dimensional Analysis.- §1. Linear operators on Hilbert spaces.- §2. Fock spaces and second quantization.- §3. Countably normed spaces and nuclear spaces.- §4. Borel measures on topological linear spaces.- II Malliavin Calculus.- §1. Gaussian probability spaces and Wiener chaos decomposition.- §2. Differential calculus of functionals, gradient and divergence operators.- §3. Meyer’s inequalities and some consequences.- §4. Densities of non-degenerate functionals.- III Stochastic Calculus of Variation for Wiener Functionals.- §1. Differential calculus of Itô functionals and regularity of heat kernels.- §2. Potential theory over Wiener spaces and quasi-sure analysis.- §3. Anticipating stochastic calculus.- IV General Theory of White Noise Analysis.- §1. General framework for white noise analysis.- §2. Characterization of functional spaces.- §3. Products and Wick products of functionals.- §4. Moment characterization of distributions and positive distributions.- V Linear Operators on Distribution Spaces.- §1. Analytic calculus for distributions.- §2. Continuous linear operators on distribution spaces.- §3. Integral kernel operators and integral kernel representation for operators.- §4. Applications to quantum physics.- Appendix A Hermite polynomials and Hermite functions.- Appendix B Locally convex spaces amd 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.- Index of Symbols.

'The book is well written and nicely structured [...] will surely become a valuable resource for specialists in stochastic analysis as well as mathematical physicists.' Mathematical Reviews (2002)