Existence of weak solutions to stochastic evolution inclusions

Jakubowski, Adam; Kamenskiı̆, Mikhail I.; Raynaud de Fitte, Paul

doi:10.1016/j.crma.2004.12.015

Probability Theory

Existence of weak solutions to stochastic evolution inclusions
[Existence de solutions faibles d'inclusions d'évolution stochastiques]

Jakubowski, Adam ¹ ; Kamenskiı̆, Mikhail I.² ; Raynaud de Fitte, Paul ³

¹ Nicholas Copernicus University, Faculty of Mathematics and Informatics, ul. Chopina 12/18, 87-100 Toruń, Poland
² Departement of Mathematics, State University of Voronezh, Voronezh, Universitetskaja pl. 1, 394693, Russia
³ Laboratoire de mathematique R. Salem, UMR CNRS 6085, UFR sciences, université de Rouen, 76821 Mont Saint Aignan cedex, France

Comptes Rendus. Mathématique, Tome 340 (2005) no. 3, pp. 229-234.

Résumé
Abstract

Nous démontrons l'existence d'une solution d'évolution faible (ou solution-mesure d'évolution) de l'inclusion différentielle stochastique dans un espace de Hilbert $d X_{t} \in A X_{t} d t + F (t, X_{t}) d t + G (t, X_{t}) d W_{t}$ où W est un mouvement brownien cylindrique, A est un opérateur linéaire qui engendre un semi-groupe de classe $C_{0}$ , F et G sont des multifonctions à valeurs convexes compactes vérifiant une condition de croissance linéaire ainsi qu'une condition plus générale que la condition de Lipschitz. La solution faible est construite au sens des mesures de Young. Lorsque F et G sont univoques, on obtient l'existence d'une solution forte.

We prove the existence of a weak mild solution (or mild solution-measure) to the Cauchy problem for the semilinear stochastic differential inclusion in a Hilbert space $d X_{t} \in A X_{t} d t + F (t, X_{t}) d t + G (t, X_{t}) d W_{t}$ where W is a cylindrical Wiener process, A is a linear operator which generates a $C_{0}$ -semigroup, F and G are multifunctions with convex compact values satisfying a linear growth condition and a condition weaker than the Lipschitz condition. The weak solution is constructed in the sense of Young measures. In the case when F and G are single-valued, we obtain the existence of a strong solution.

Reçu le : 2004-07-21
Accepté le : 2004-12-14
Publié le : 2005-01-07

DOI : 10.1016/j.crma.2004.12.015

Affiliations des auteurs :

Jakubowski, Adam ¹ ; Kamenskiı̆, Mikhail I. ² ; Raynaud de Fitte, Paul ³

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     title = {Existence of weak solutions to stochastic evolution inclusions},
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     pages = {229--234},
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Jakubowski, Adam; Kamenskiı̆, Mikhail I.; Raynaud de Fitte, Paul. Existence of weak solutions to stochastic evolution inclusions. Comptes Rendus. Mathématique, Tome 340 (2005) no. 3, pp. 229-234. doi : 10.1016/j.crma.2004.12.015. http://www.numdam.org/articles/10.1016/j.crma.2004.12.015/

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