The aim of this book is to give a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. These are a generalization of stochastic differential equations as introduced by Itô and Gikham that occur, for instance, when describing random phenomena that crop up in science and engineering, as well as in the study of differential equations. The book is divided into three parts. In the first the authors give a self-contained exposition of the basic properties of probability measure on separable Banach and Hilbert spaces, as required later; they assume a reasonable background in probability theory and finite dimensional stochastic processes. The second part is devoted to the existence and uniqueness of solutions of a general stochastic evolution equation, and the third concerns the qualitative properties of those solutions. Appendices gather together background results from analysis that are otherwise hard to find under one roof. The book ends with a comprehensive bibliography that will contribute to the book's value for all working in stochastic differential equations.

This book is devoted to the 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; invariant measures for specific models. The focus is on models of dynamical processes affected by white noise, which are described by partial differential equations such as the reaction-diffusion equations or Navier–Stokes equations. Besides existence and uniqueness questions, special attention is paid to the asymptotic behaviour of the solutions, to invariant measures and ergodicity. Some of the results found here are presented for the first time. For all whose research interests involve stochastic modelling, dynamical systems, or ergodic theory, this book will be an essential purchase.

Parti Foundations 1 1 Why equations with Levy noise? 3 1.1 Discrete-time dynamical systems 3 1.2 Deterministic continuous-time systems ' ' ' 5 1.3 Stochastic continuous-time systems 6 1.4 Courrege's theorem 8 1.5 Ito's approach 9 1.6 Infinite-dimensional case 12 2 Analytic preliminaries 13 2.1 Notation ' 13 2.2 Sobolev and Holder spaces 13 2.3 L p-and C p-spaces " ' ' 15 2.4 Lipschitz functions and composition operators 16 2.5 Differential operators 17 3 Probabilistic preliminaries 20 "3.1 Basic definitions ~ 20 3.2 Kolmogorov existence theorem 22 3.3 Random elements in Banacti spaces 23 3.4 Stochastic processes in Banach spaces 25 3.5 Gaussian measures on Hilbert spaces 28 3.6 Gaussian 1 measures on topological spaces 30 ' 3.7 Submartingales , ; •. 31 3.8 Semimartingales • 36 3.9 Burkholder-Davies-Guhdy inequalities .

This chapter is devoted to some basic results on Gaussian measures on separable Hilbert spaces, including the Cameron-Martin and Feldman-Hajek formulae. The greater part of the results are presented with complete proofs.

Now in its second edition, this book gives a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. In the first part the authors give a self-contained exposition of the basic properties of probability measure on separable Banach and Hilbert spaces, as required later; they assume a reasonable background in probability theory and finite dimensional stochastic processes. The second part is devoted to the existence and uniqueness of solutions of a general stochastic evolution equation, and the third concerns the qualitative properties of those solutions. Appendices gather together background results from analysis that are otherwise hard to find under one roof. This revised edition includes two brand new chapters surveying recent developments in the area and an even more comprehensive bibliography, making this book an essential and up-to-date resource for all those working in stochastic differential equations.