Ergodicity for infinite dimensional systems.

*(English)*Zbl 0849.60052
London Mathematical Society Lecture Note Series. 229. Cambridge: Cambridge Univ. Press. xi, 339 p. (1996).

The main object of this monograph is to provide sufficient criteria for the existence of an invariant probability measure for a large class of stochastic evolution equations on a Banach or Hilbert space. The first four chapters contain an introduction to ergodic theory, Markovian semigroups, mixing properties and the Feller property. They include for example proofs of the Koopman-von Neumann theorem, the Krylov-Bogolyubov existence theorem, strong laws and Doob’s theorem on regular Markovian semigroups. This part is largely encyclopedic with hardly any examples.

Stochastic differential equations (SDEs) driven by a (cylindrical) Wiener process are introduced in Chapter 5. For the definition of the stochastic integral and for proofs of fundamental inequalities the reader is referred to the authors’ monograph “Stochastic equations in infinite dimensions” (1992; Zbl 0761.60052). They provide (a sketch of the) proofs of theorems on the existence and uniqueness of mild solutions for certain SDEs however. Criteria for the existence and uniqueness of invariant measures are given in Chapters 6 and 7 and are applied – sometimes with appropriate modifications – to special examples like stochastic delay equations, reaction-diffusion equations, spin systems, Burgers equation and Navier-Stokes equations in the third part of the book. Sufficient conditions for the existence of invariant measures are formulated in terms of dissipativity properties of the drift. Chapter 7 provides conditions for the strong Feller property and irreducibility which together imply uniqueness of an invariant measure. The authors mention the possibility to obtain sufficient conditions for existence resp. uniqueness via Lyapunov functions resp. coupling but refer the reader to the literature for corresponding results.

The book is written for readers with a solid background in measure theory, functional analysis, probability and stochastic analysis. Terms like “\(\pi\)-system”, “Kolmogorov’s extension theorem”, “progressive measurability”, “trace class operator” are used without explanation. Readers who have this background and who are interested in the asymptotics of infinite-dimensional stochastic systems are likely to find the book well-written and very useful both because of the abstract results and the applications.

Stochastic differential equations (SDEs) driven by a (cylindrical) Wiener process are introduced in Chapter 5. For the definition of the stochastic integral and for proofs of fundamental inequalities the reader is referred to the authors’ monograph “Stochastic equations in infinite dimensions” (1992; Zbl 0761.60052). They provide (a sketch of the) proofs of theorems on the existence and uniqueness of mild solutions for certain SDEs however. Criteria for the existence and uniqueness of invariant measures are given in Chapters 6 and 7 and are applied – sometimes with appropriate modifications – to special examples like stochastic delay equations, reaction-diffusion equations, spin systems, Burgers equation and Navier-Stokes equations in the third part of the book. Sufficient conditions for the existence of invariant measures are formulated in terms of dissipativity properties of the drift. Chapter 7 provides conditions for the strong Feller property and irreducibility which together imply uniqueness of an invariant measure. The authors mention the possibility to obtain sufficient conditions for existence resp. uniqueness via Lyapunov functions resp. coupling but refer the reader to the literature for corresponding results.

The book is written for readers with a solid background in measure theory, functional analysis, probability and stochastic analysis. Terms like “\(\pi\)-system”, “Kolmogorov’s extension theorem”, “progressive measurability”, “trace class operator” are used without explanation. Readers who have this background and who are interested in the asymptotics of infinite-dimensional stochastic systems are likely to find the book well-written and very useful both because of the abstract results and the applications.

Reviewer: M.Scheutzow (Berlin)