Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Estimate for $P_{t}D$ for the stochastic Burgers equation

Giuseppe Da Prato and Arnaud Debussche

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Abstract

We consider the Burgers equation on $H=L^{2}(0,1)$ perturbed by white noise and the corresponding transition semigroup $P_{t}$. We prove a new formula for $P_{t}D_{x}\varphi$ which depends on $\varphi$ but not on its derivative. This formula allows us to provide a bound on $D_{x}\varphi$ in $L^{2}(H,\nu)$ where $\nu$ is the invariant measure of $P_{t}$. Some new consequences for the invariant measure $\nu$ of $P_{t}$ are discussed as its Fomin differentiability and an integration by parts formula which generalises the classical one for Gaussian measures.

Résumé

Nous considèrons l’équation de Burgers stochastique sur $H=L^{2}(0,1)$ dirigée par un bruit blanc, de semi-groupe de transition $P_{t}$, et démontrons une nouvelle formule qui permet d’exprimer $P_{t}D_{x}\varphi$ en terme de $\varphi$ mais pas de sa différentielle. Celle-ci nous permet d’obtenir des estimations sur $D_{x}\varphi$ dans $L^{2}(H,\nu)$, où $\nu$ est la mesure invariante de $P_{t}$, dont découlent quelques conséquences telles que l’existence de dérivées de Fomin pour $\nu$ ou encore une formule d’intégration par partie qui généralise celle bien connue pour les mesures gaussiennes.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1248-1258.

Dates
Received: 13 November 2014
Revised: 21 April 2015
Accepted: 21 April 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723519

Digital Object Identifier
doi:10.1214/15-AIHP685

Mathematical Reviews number (MathSciNet)
MR3531708

Zentralblatt MATH identifier
1350.60057

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25]

Keywords
Stochastic Burgers equation invariant measure Fomin differentiability

Citation

Da Prato, Giuseppe; Debussche, Arnaud. Estimate for $P_{t}D$ for the stochastic Burgers equation. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1248--1258. doi:10.1214/15-AIHP685. https://projecteuclid.org/euclid.aihp/1469723519


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