Invariant measures for dichotomous stochastic differential equations in Hilbert spaces
[Mesures invariantes pour des équations différentielles stochastiques à dichotomies exponentielles dans les espaces de Hilbert]
Van Gaans, Onno1 ; Verduyn Lunel, Sjoerd2
1 Department of Applied Mathematical Analysis, Faculty ITS, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands
2 Mathematical Institute, Leiden University, PO Box 9512, 2300 RA Leiden, The Netherlands
Comptes Rendus. Mathématique, Tome 334 (2002) no. 12, pp. 1083-1088.

Nous étudions l'existence de mesures invariantes pour des équations différentielles stochastiques semilinéaires dans les espaces de Hilbert. Nous considérons des bruits de dimension infinie qui sont blancs en la variable du temps et colorés en les variables de l'espace et nous supposons que les nonlinéarités sont lipschitziennes. Supposons en outre que l'équation a une dichotomie au sens où le semigroupe engendré par la partie linéaire est hyperbolique et les constantes lipschitziennes ne sont pas trop grandes. Nous démontrons alors que l'existence d'une solution à variance bornée entraîne l'existence d'une mesure invariante.

We study existence of invariant measures for semilinear stochastic differential equations in Hilbert spaces. We consider infinite dimensional noise that is white in time and colored in space and we assume that the nonlinearities are Lipschitz continuous. We show that if the equation is dichotomous in the sense that the semigroup generated by the linear part is hyperbolic and the Lipschitz constants of the nonlinearities are not too large, then existence of a solution with bounded mean squares implies existence of an invariant measure.

Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02410-X
Van Gaans, Onno 1 ; Verduyn Lunel, Sjoerd 2

1 Department of Applied Mathematical Analysis, Faculty ITS, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands
2 Mathematical Institute, Leiden University, PO Box 9512, 2300 RA Leiden, The Netherlands 
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Van Gaans, Onno; Verduyn Lunel, Sjoerd. Invariant measures for dichotomous stochastic differential equations in Hilbert spaces. Comptes Rendus. Mathématique, Tome 334 (2002) no. 12, pp. 1083-1088. doi : 10.1016/S1631-073X(02)02410-X. http://www.numdam.org/articles/10.1016/S1631-073X(02)02410-X/

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