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.

INTRODUCTION

Purpose. The purpose of this report is to provide a multi-type extension, allowing weak interaction in both space and type, of the well-known results of Dawson and Gärtner (1987) [DG] regarding large deviations from the McKean-Vlasov limit for weakly interacting diffusions. This is achieved byintegrating more systematically the previous work Djehiche and Kaj (1994) [DK], in which a large deviation result is derived for a class of measure-valued jump processes, with the setting of the Dawson-Gärtner large deviation principle. Necessarilly, some aspects of such an extension will be mere notational rather than substantial. We will try to focus on those parts that are less evident and to point out some techniques from [DK] which can be used as an alternative to those of [DG].

INTRODUCTION

The definition of several types of stochastic integrals for anticipating integrands put the basis for the development, in recent years, of an anticipating stochastic calculus. It is natural to consider, as an application, some problems that can be stated formally as stochastic differential equations, but that cannot have a sense within the theory of non-anticipating stochastic integrals. For example, this is the case if we impose to an s.d.e. an initial condition which is not independent of the driving process, or if we prescribe boundary conditions for the solution.

In this paper, we will try to survey the work already done concerning s.d.e. with boundary conditions, and to explain in some detail a method based in transformations and change of measure in Wiener space. An alternative approach is sketched in the last Section.

Transformations on Wiener space provide a natural method, among others, to prove existence and uniqueness results for nonlinear equations. At the same time, a Girsanov type theorem for not necessarily adapted transformations allows to study properties of the laws of the solutions from properties of the solution to an associated linear equation. A natural first question about these laws is to decide if they satisfy some kind of Markov (or conditional independence) property.

In Section 2, we introduce stochastic differential equations with boundary conditions, the particular instances that have been studied, and the kind of results obtained concerning conditional independence properties of the solutions. The short Section 3 outlines the idea of the method of transformations and change of measure. In Section 4, we recall the necessary elements of Wiener space analysis in order to enounce the Girsanov type theorem we want to apply.