Preface

Summary

This book is devoted to asymptotic properties of solutions of stochastic evolution equations in infinite dimensional spaces. It is divided into three parts: Markovian Dynamical Systems, Invariant Measures for Stochastic Evolution Equations, and Invariant Measures for Specific Models.

In the first part of the book we recall basic concepts of the theory of dynamical systems and we link them with the theory of Markov processes. In this way such notions as ergodic, mixing, strongly mixing Markov processes will be special cases of well known concepts of a more general theory. We also give a proof of the Koopman–von Neumann ergodic theorem and, following Doob, we apply it in Chapter 4 to a class of regular Markov processes important in applications. We also include the Krylov–Bogoliubov theorem on existence of invariant measures and give a semigroup characterization of ergodic and mixing measures.

The second part of the book is concerned with invariant measures for important classes of stochastic evolution equations. The main aim is to formulate sufficient conditions for existence and uniqueness of invariant measures in terms of the coefficients of the equations.

We develop first two methods for establishing existence of invariant measures exploiting either compactness or dissipativity properties of the drift part of the equation. We also give necessary and sufficient conditions for existence of invariant measures for general linear systems.