Kolmogorov Equations for Stochastic PDEs - Prato, Giuseppe (Birkhäuser Basel, 2004)
This book is devoted to some basic stochastic partial differential equations, in particular reaction-diffusion equations, Burgers and Navier-Stokes equations per turbed by noise. Particular attention is paid to the corresponding Kolmogorov equations which are elliptic or parabolic equations with infinitely many variables. The aim of the book is to present the basic elements of stochastic PDEs in a simple and self-contained way in order to cover the program of one year PhD course both in Mathematics and in Physics. The needed prerequisites are some basic knowledge of probability, functional analysis (including fundamental properties of Gaussian measures) and partial dif ferential equations. This book is an expansion of a course given by the author in 1997 at the "Center de Recerca Matematica" in Barcelona (see [30]), which I thank for the warm hospitality. I wish also to thank B. Goldys for reading the manuscript and making several useful comments. This work was also supported by the research program "Analisi e controllo di equazioni di evoluzione deterministiche e stocastiche" from the Italian "Ministero della Ricerca Scientifica e Tecnologica" . Pisa, October 2004 Giuseppe Da Prato Chapter 1 Introduction and Preliminaries 1.1 Introduction We are here concerned with a stochastic differential equation in a separable Hilbert space H, dX(t,x) = (AX(t, x) + F(X(t, x)))dt + B dW(t), t > 0, x E H, { (1.1) X(O,x) = x, x E H.