## The Annals of Probability

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

#### 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 nonlocally 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.

#### Article information

Source
Ann. Probab., Volume 34, Number 2 (2006), 663-727.

Dates
First available in Project Euclid: 9 May 2006

https://projecteuclid.org/euclid.aop/1147179986

Digital Object Identifier
doi:10.1214/009117905000000666

Mathematical Reviews number (MathSciNet)
MR2223955

Zentralblatt MATH identifier
1106.35127

#### Citation

Röckner, Michael; Sobol, Zeev. Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations. Ann. Probab. 34 (2006), no. 2, 663--727. doi:10.1214/009117905000000666. https://projecteuclid.org/euclid.aop/1147179986

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