Stochastic equations in infinite dimensions. 2nd ed.

*(English)*Zbl 1317.60077
Encyclopedia of Mathematics and Its Applications 152. Cambridge: Cambridge University Press (ISBN 978-1-107-05584-1/hbk; 978-1-107-29551-3/ebook). xviii, 493 p. (2014).

The book under review is an update and improvement of the first edition as of 1992. It is a systematic and a self-contained introduction to the theory of semilinear stochastic evolution equations of the form
\[
dX(t)=[AX(t)+F(t,X(t))]\,dt+G(t,X(t))\,dW(t)
\]
in Hilbert and Banach spaces where \(A\) is a linear (unbounded) operator, \(F\) and \(G\) are given non-linearities and \(W\) is an infinite-dimensional Wiener process.

The book opens by several motivating examples of stochastic equations arising in physics, neurophysiology, population genetics and finance.

In Part I, the authors recall, to a necessary extent, basic properties of vector-valued random variables and probability measures on Polish spaces and basics of the stochastic analysis in infinite dimensions including the construction of the stochastic integral with respect to an infinite-dimensional Wiener process.

Part II addresses problems of existence, uniqueness and regularity of weak, strong, mild and martingale solutions in a systematic exposition, including a detailed presentation of the factorization method.

Part III deals with qualitative properties of solutions such as the Markov and the strong Markov property, the dependence of solutions on initial data, the large time behavior (including a study of invariant measures and the mixing and the recurrence properties) or the small noise asymptotic behavior (including a presentation of the large deviation principle and the exit problem). Also, the role of the deterministic Kolmogorov equation is explained and the Girsanov theorem is presented.

The second edition of the book contains newly two chapters 13 and 14 surveying significant results concerning many important specific equations and recent developments in the field, respectively, and an updated and comprehensive bibliography.

The book opens by several motivating examples of stochastic equations arising in physics, neurophysiology, population genetics and finance.

In Part I, the authors recall, to a necessary extent, basic properties of vector-valued random variables and probability measures on Polish spaces and basics of the stochastic analysis in infinite dimensions including the construction of the stochastic integral with respect to an infinite-dimensional Wiener process.

Part II addresses problems of existence, uniqueness and regularity of weak, strong, mild and martingale solutions in a systematic exposition, including a detailed presentation of the factorization method.

Part III deals with qualitative properties of solutions such as the Markov and the strong Markov property, the dependence of solutions on initial data, the large time behavior (including a study of invariant measures and the mixing and the recurrence properties) or the small noise asymptotic behavior (including a presentation of the large deviation principle and the exit problem). Also, the role of the deterministic Kolmogorov equation is explained and the Girsanov theorem is presented.

The second edition of the book contains newly two chapters 13 and 14 surveying significant results concerning many important specific equations and recent developments in the field, respectively, and an updated and comprehensive bibliography.

Reviewer: Martin Ondreját (Praha)