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 GinzburgLandau, the
generalized stochastic Burgers, and the stochastic NavierStokes 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.
