Stochastic Partial Differential Equations and Invariant Measures

We shall first review the connection between ordinary stochastic differential equations (SODE) on R^d and parabolic partial differential equations, more precisely the associated Kolmogorov equations. The point will be that the latter can be viewed as a linearization of the first. Subsequently, we shall illustrate this connection also for SODE's in infinite dimensions. A purely analytic approach to solve the corresponding Kolmogorov equations in infinitely many variables will be presented. As a consequence one obtains weak solutions for the SODE. Applications include the case where the infinite dimensional SODE is a parabolic stochastic partial differential equation, as e.g. the stochastic Ginzburg-Landau, the generalized stochastic Burgers, and the stochastic Navier-Stokes equation. In this talk we shall particularly concentrate on applying the results to the stochastic porous media equation. In particular,we shall prove the existence and uniqueness of an infinitesimally invariant measure in this case.