Preface

Summary

This book is devoted to stochastic evolution equations on infinite dimensional spaces, mainly Hilbert and Banach spaces. These equations are generalizations of Itô stochastic equations introduced in the 1940s by Itô [423] and in a different form by Gikhman [347].

First results on infinite dimensional Itô equations started to appear in the mid-1960s and were motivated by the internal development of analysis and the theory of stochastic processes on the one hand, and by a need to describe random phenomena studied in the natural sciences like physics, chemistry, biology, engineering as well as in finance, on the other hand.

Hilbert space valued Wiener processes and, more generally, Hilbert space valued diffusion processes, were introduced by Gross [363] and Daleckii [183] as a tool to investigate the Dirichlet problem and some classes of parabolic equations for functions of infinitely many variables. An infinite dimensional version of anOrnstein–Uhlenbeck process was introduced by Malliavin [518, 519] as a tool for stochastic study of the regularity of fundamental solutions of deterministic parabolic equations.

Stochastic parabolic type equations appeared naturally in the study of conditional distributions of finite dimensional processes in the form of the so called nonlinear filtering equation derived by Fujisaki, Kallianpur and Kunita [330] and Liptser and Shiryayev [501] or as a linear stochastic equation introduced by Zakaï [737]. Another source of inspiration was provided by the study of stochastic flows defined by ordinary stochastic equations. Such flows are in fact processes with values in an infinite dimensional space of continuous or even more regular mappings acting in a Euclidean space.