The Annals of Probability

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

Michael Röckner and Zeev Sobol

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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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1147179986

Digital Object Identifier
doi:10.1214/009117905000000666

Mathematical Reviews number (MathSciNet)
MR2223955

Zentralblatt MATH identifier
1106.35127

Subjects
Primary: 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65]
Secondary: 35J70: Degenerate elliptic equations 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
Stochastic Burgers equation Kolmogorov equation infinite-dimensional background space weighted space of continuous functions Lyapunov function Feller semigroup diffusion process

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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