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Kolmogorov equations in infinite dimensions: well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations. (English) Zbl 1106.35127

The paper develops a new method to uniquely solve a large class of heat equations \[ \frac{du}{dt}=Lu \] in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions on the underlying infinite-dimensional Banach space weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts.
Apart from general analytic interest, the main motivation of the authors was to apply this to uniquely solve martingale problems in the sense of Strook-Varadan given by SPDEs 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.

MSC:

35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
60J60 Diffusion processes
35Q53 KdV equations (Korteweg-de Vries equations)
35J70 Degenerate elliptic equations
47D06 One-parameter semigroups and linear evolution equations
47D08 Schrödinger and Feynman-Kac semigroups
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents
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