Geodesic
Probability Theory
Invariant measures of stochastic partial differential equations and conditioned diffusions
[Mesures invariantes d'équations aux dérivées partielles stochastiques et diffusions conditionées]

Présenté par : Paul Malliavin

Reznikoff, Maria G.1 ; Vanden-Eijnden, Eric2
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, Tome 340 (2005) no. 4, pp. 305-308.

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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 ut=uxx+f(u)+2ɛη(x,t), where η(x,t) is a space–time white-noise, is identical to the law of the bridge process associated to dU=a(U)dx+ɛdW(x), provided that a and f are related by ɛa″(u)+2a′(u)a(u)=−2f(u), u∈R. Some consequences of this connection are investigated, including the existence and properties of the invariant measure for the SPDE on the line, x∈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 ut=uxx+f(u)+2ɛη(x,t), où η(x,t) est un bruit blanc spatio-temporel, est la même que la loi du processus pont associé à dU=a(U)dx+ɛdW(x), pourvu que a et f soient reliés comme ɛa″(u)+2a′(u)a(u)=−2f(u), u∈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∈R.

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Accepté le :
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DOI : 10.1016/j.crma.2004.12.025

Reznikoff, Maria G. 1 ; Vanden-Eijnden, Eric 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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Reznikoff, Maria G.; Vanden-Eijnden, Eric. Invariant measures of stochastic partial differential equations and conditioned diffusions. Comptes Rendus. Mathématique, Tome 340 (2005) no. 4, pp. 305-308. doi : 10.1016/j.crma.2004.12.025. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2004.12.025/

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