Kolgomorov Equations for Stochastic PDEs
This monograph treats the stochastic differential equation dX(t,x) = AX(t,x) + F(X(t,x)) dt + B dW, X(0,x) = x, x є H in a separable Hilbert space H, where A and B are operators, F is a nonlinear map and W is the cylindrical Wiener process. A solution X(t,x) is required to be an L2-continuous stochastic process adapted to W(t). The equation is known as a model for the evolution of an infinite dimensional dynamical system that is perturbed by noise and covers reaction-diffusion, Burges and Navier-Stokes equations. The corresponding elliptic and parabolic Kolmogorov partial differential equations are discussed with considerable details and in each case the transition group, the strong Feller property, irreducibility and invariant measures are investigated. This is a good book for study; it is a clear and compact presentation of a topic that is of considerable interest both in mathematics and statistical physics. Some results that appear here are completely new. The text could provide material for an advanced course for students who are familiar with stochastic analysis, basic functional analysis and the theory of partial differential equations.