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. In 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 Note this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.
Nous proposons une nouvelle méthode de résoudre une large classe d'équations de chaleur, c'est-à-dire, d'équations de Kolmogorov en dimension infinie. Nous considèrons ces équations dans les espaces des fonctions faiblement séquentiellement continues et subordonnées aux fonctions du type de Liapounoff appropriées. Nos résultats donnent la première construction d'une solution qui existe partout dans le cas de coefficients non lipschitziens. Ces études sont motivées par des applications aux problèmes de martingales au sens de Stroock–Varadhan pour les équations stochastiques aux dérivées partielles de l'hydrodynamique du type de Navier–Stokes. En particulier, l'équation stochastique de Burgers est analysée. L'unicité est établie au sens des flots markoviens.
Accepted:
Published online:
Michael Röckner 1; Zeev Sobol 2
@article{CRMATH_2004__338_12_945_0,
author = {Michael R\"ockner and Zeev Sobol},
title = {A new approach to {Kolmogorov} equations in infinite dimensions and applications to stochastic generalized {Burgers} equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {945--949},
publisher = {Elsevier},
volume = {338},
number = {12},
year = {2004},
doi = {10.1016/j.crma.2004.03.024},
language = {en},
}
TY - JOUR AU - Michael Röckner AU - Zeev Sobol TI - A new approach to Kolmogorov equations in infinite dimensions and applications to stochastic generalized Burgers equations JO - Comptes Rendus. Mathématique PY - 2004 SP - 945 EP - 949 VL - 338 IS - 12 PB - Elsevier DO - 10.1016/j.crma.2004.03.024 LA - en ID - CRMATH_2004__338_12_945_0 ER -
%0 Journal Article %A Michael Röckner %A Zeev Sobol %T A new approach to Kolmogorov equations in infinite dimensions and applications to stochastic generalized Burgers equations %J Comptes Rendus. Mathématique %D 2004 %P 945-949 %V 338 %N 12 %I Elsevier %R 10.1016/j.crma.2004.03.024 %G en %F CRMATH_2004__338_12_945_0
Michael Röckner; Zeev Sobol. A new approach to Kolmogorov equations in infinite dimensions and applications to stochastic generalized Burgers equations. Comptes Rendus. Mathématique, Volume 338 (2004) no. 12, pp. 945-949. doi : 10.1016/j.crma.2004.03.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.03.024/
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