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.

The example after Theorem 3.2 showed that for a continuous distribution function F such as for U[0, 1], the set of all possible functions √n(Fn − F), even for n = 1, is nonseparable in the sup norm, and all its subsets are closed, including those corresponding to nonmeasurable sets of possible values of the observation X1. Therefore, the classical definition of convergence in law, or weak convergence, which works in separable metric spaces, does not work in this case, So, in Chapter 3, functions f* and upper expectations E* were used to get around measurability problems.

But, in the classical Glivenko–Cantelli theorem, saying that supx |(Fn − F)(x)| → 0 almost surely as n → ∞ for any distribution function F on ℝ and its empirical distribution functions Fn (RAP, Theorem 11.4.2), there is no measurability problem. The supremum is measurable, as it can be restricted to rational x by right-continuity of Fn and F. The collection C of left half-lines (−∞, x] is linearly ordered by inclusion and so has S(C) = 1, and for it, not only the Glivenko–Cantelli theorem but, after suitable formulations (Theorem 1.8 or, less specifically, Chapter 3), the uniform central limit theorem (Donsker property) holds for any probability measure P on the Borel sets of ℝ.