This book is devoted to stochastic evolution equations on infinite dimensional spaces, mainly Hilbert and Banach spaces. These equations are generalizations of Ito stochastic equations introduced in the 1940s by Ito [127] and in a different form by Gikhman [100].

First results on infinite dimensional Ito's equations started to appear in the mid 1960s and were motivated by internal development of analysis and theory of stochastic processes on one side, and by a need to describe random phenomena studied in natural sciences like physics, chemistry, biology, and in control theory, on the other side. Hilbert space valued Wiener processes and, more generally, Hilbert space valued diffusion processes, were introduced by Gross [105] and Daleckii [47] as a tool to investigate Dirichlet problems and some classes of parabolic equations for functions which depend on infinitely many variables. An infinite dimensional version of an Ornstein-Uhlenbeck process was introduced by P. Malliavin [168] for a stochastic study of regularity of fundamental solutions of deterministic parabolic equations (see also Stroock [218]). Stochastic parabolic type equations have appeared naturally in the study of conditional distributions of finite dimensional processes in the form of the socalled non-linear filtering equation derived by Fujisaki, Kallianpur and Kunita [94] and Liptzer, Shiryayev [163] or as a linear stochastic equation introduced by Zakai [252]. Another source of inspiration was provided by 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 Euclidian space.