Existence of weak solutions to stochastic evolution inclusions

Adam Jakubowski (1) , Mikhail Kamenskii (2) , Paul Raynaud de Fitte (3)

1 UMK - Faculty of Mathematics and Informatics [ Toruń]
2 VSU - Departement of Mathematics
3 LMRS - Laboratoire de Mathématiques Raphaël Salem

We consider the Cauchy problem for a semilinear stochastic differential inclusion in a Hilbert space. The linear operator generates a strongly continuous semigroup and the nonlinear term is multivalued and satisfies a condition which is more heneral than the Lipschitz condition. We prove the existence of a mild solution to this problem. This solution is not "strong" in the probabilistic sense, that is, it is not defined on the underlying probability space, but on a larger one, which provides a "very good extension" in the sense of Jacod and Mémin. Actually, we construct this solution as a Young measure, limit of approximated solutions provided by the Euler scheme. The compactness in the space of Young measures of this sequence of approximated solutions is obtained by proving that some measure of noncompactness equals zero.

Mots clés: measure of noncompactness; solution-measure; very good extension; weak solution; Young measure; semigroup; semilinear; multifunction

Domaines: Probabilités [math.PR]

Adam Jakubowski, Mikhail Kamenskii, Paul Raynaud de Fitte. Existence of weak solutions to stochastic evolution inclusions. Stochastic Analysis and Applications, 2005, 23 (4), pp.723-749. ⟨10.1081/SAP-200064469⟩. ⟨hal-00002365v2⟩