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

Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations