Invariant measures of stochastic partial differential equations and conditioned diffusions

Maria G. Reznikoff; Eric Vanden-Eijnden

doi:10.1016/j.crma.2004.12.025

Probability Theory

Invariant measures of stochastic partial differential equations and conditioned diffusions

Presented by: Paul Malliavin

Maria G. Reznikoff ¹; Eric Vanden-Eijnden ²

¹ Institute for Applied Mathematics, University of Bonn, Wegelerstraße 10, 53115 Bonn, Germany
² Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 305-308.

Abstract
Résumé

This work establishes and exploits a connection between the invariant measure of stochastic partial differential equations (SPDEs) and the law of bridge processes. Namely, it is shown that the invariant measure of $u_{t} = u_{x x} + f (u) + \sqrt{2 ɛ} η (x, t)$ , where $η (x, t)$ is a space–time white-noise, is identical to the law of the bridge process associated to $d U = a (U) d x + \sqrt{ɛ} d W (x)$ , provided that a and f are related by $ɛ a^{″} (u) + 2 a^{'} (u) a (u) = - 2 f (u)$ , $u \in R$ . Some consequences of this connection are investigated, including the existence and properties of the invariant measure for the SPDE on the line, $x \in R$ .

On montre et exploite une connection entre la mesure invariante d'équations aux dérivées partielles stochastiques et les lois de processus ponts. En l'occurence, on montre que la mesure invariante de $u_{t} = u_{x x} + f (u) + \sqrt{2 ɛ} η (x, t)$ , où $η (x, t)$ est un bruit blanc spatio-temporel, est la même que la loi du processus pont associé à $d U = a (U) d x + \sqrt{ɛ} d W (x)$ , pourvu que a et f soient reliés comme $ɛ a^{″} (u) + 2 a^{'} (u) a (u) = - 2 f (u)$ , $u \in R$ . Quelques conséquences de cette connection sont étudiées, comme l'existence et les propriétés d'une mesure invariante de l'équations aux dérivées partielle stochastique sur la ligne, $x \in R$ .

Received: 2004-09-29
Accepted: 2004-12-14
Published online: 2005-02-02

DOI: 10.1016/j.crma.2004.12.025

Author's affiliations:

Maria G. Reznikoff ¹; Eric Vanden-Eijnden ²

¹ Institute for Applied Mathematics, University of Bonn, Wegelerstraße 10, 53115 Bonn, Germany
² Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

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     author = {Maria G. Reznikoff and Eric Vanden-Eijnden},
     title = {Invariant measures of stochastic partial differential equations and conditioned diffusions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {305--308},
     publisher = {Elsevier},
     volume = {340},
     number = {4},
     year = {2005},
     doi = {10.1016/j.crma.2004.12.025},
     language = {en},
}

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Maria G. Reznikoff; Eric Vanden-Eijnden. Invariant measures of stochastic partial differential equations and conditioned diffusions. Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 305-308. doi : 10.1016/j.crma.2004.12.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.025/

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