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Kolmogorov equations for stochastic PDEs. (English) Zbl 1066.60061

The author is concerned with the transition semigroup of stochastic partial differential equations in a Hilbert space \(H\) of the form \[ dX(t,x)=(AX(t,x)+F(X(t,x)))dt+BdW(t), \quad t>0,\quad x\in H, \]
\[ X(0,x)=x,\quad x\in H, \] where \(A:D(A)\subset H\rightarrow H\) is the infinitesimal generator of a strongly continuous semigroup in \(H\), \(B\) is a bounded operator from another Hilbert space \(U\), \(F:D(F)\subset H\rightarrow H\) is a non-linear mapping and \(W(t)\) is a cylindrical Wiener process in \(U\). Solutions of this equation are considered in the “mild” sense. Several properties of the transition semigroup are studied, such as Feller and strong Feller, irreducibility, existence and uniqueness of invariant measures and particular attention is paid to Kolmogorov equations. The book aims to be self-contained and cover a one year PhD course both in Mathematics and Physics. The contents are the following:
Chapter 1 is devoted to an introduction and some preliminaries needed along the book. In Chapter 2 the author deals with the above equation when the nonlinearity \(F\) vanishes; the Ornstein-Uhlenbeck semigroup is defined and studied, proving the “carré du champs” identity and the Poincaré and log-Sobolev inequalities [most part of this chapter is based on the author and L. Tubaro, Czech. Math. J. 51, No. 4, 685–699 (2001; Zbl 0996.47028) and the author and J. Zabczyk, “Ergodicity for infinite dimensional systems” (1996; Zbl 0849.60052)]. Chapter 3 is devoted to study stochastic differential equations with Lipschitz nonlinearities; the author analyses existence and uniqueness of “mild” solution, and properties of the transition semigroup are studied, stating also comparison results for the invariant measure in the case \(F=0\). Chapters 4, 5 and 6 are concerned with reaction-diffusion equations, the stochastic Burgers equation and the stochastic 2D Navier-Stokes equation, respectively; in these chapters the equations are solved, as well as the corresponding Kolmogorov equations, and the transition semigroups are studied.
The monograph is well written and provides a suitable tool to get involved in the semigroup approach to stochastic partial differential equations.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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