Introduction to Infinite Dimensional Stochastic Analysis

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 math ematical 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 partial differential equation for transition probabilities of Markov processes order 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 function als of Brownian paths (i. e. the Wiener functionals).

Klappentext

This book offers a concise introduction to the rapidly expanding field of infinite dimensional stochastic analysis. It treats Malliavin calculus and white noise analysis in a single book, presenting these two different areas in a unified setting of Gaussian probability spaces. Topics include recent results and developments in the areas of quasi-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. br/ emAudience:/em This volume will be of interest to researchers and graduate students whose work involves probability theory, stochastic processes, functional analysis, operator theory, mathematics of physics and abstract harmonic analysis.

Inhalt
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.