Abstract. We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly non-locally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of Stroock-Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier-Stokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.

Abstract. We establish the maximal regularity for nonautonomous Ornstein-Uhlenbeck operators in L p-spaces with respect to a family of invariant measures, where p ∈ (1, +∞). This result follows from the maximal L p-regularity for a class of elliptic operators with unbounded, time-dependent drift coefficients and potentials acting on L p (R N) with Lebesgue measure. 1.

The main purpose of this paper is to show that the generalized Schrödinger operator A V f = 1 2∆f + b∇f − V f, f ∈ C ∞ 0 (R d), is a pre-generator for which we can prove its L ∞ ` R d, dx ´-uniqueness. Moreover, we prove the L 1 (R d, dx)-uniqueness of weak solutions for the Fokker-Planck equation associated with this pre-generator. Key Words: C0-semigroups; L ∞-uniqueness; generalized Schrödinger operators; Fokker-Planck equation.