Existence of weak solutions to stochastic evolution inclusions

Abstract

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 ∈ 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. To cite this article: A. Jakubowski et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).