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Full Description
In this revised and extended version of his course notes from a 1-year course at Scuola Normale Superiore, Pisa, the author provides an introduction - for an audience knowing basic functional analysis and measure theory but not necessarily probability theory - to analysis in a separable Hilbert space of infinite dimension.
Starting from the definition of Gaussian measures in Hilbert spaces, concepts such as the Cameron-Martin formula, Brownian motion and Wiener integral are introduced in a simple way. These concepts are then used to illustrate some basic stochastic dynamical systems (including dissipative nonlinearities) and Markov semi-groups, paying special attention to their long-time behavior: ergodicity, invariant measure. Here fundamental results like the theorems of Prokhorov, Von Neumann, Krylov-Bogoliubov and Khas'minski are proved. The last chapter is devoted to gradient systems and their asymptotic behavior.
Contents
Gaussian measures in Hilbert spaces.- The Cameron-Martin formula.- Brownian motion.- Stochastic perturbations of a dynamical system.- Invariant measures for Markov semigroups.- Weak convergence of measures.- Existence and uniqueness of invariant measures.- Examples of Markov semigroups.- L2 spaces with respect to a Gaussian measure.- Sobolev spaces for a Gaussian measure.- Gradient systems.
