Stochastic Functional Partial Differential Equations: Existence, Uniqueness and Asymptotic Decay Property

Abstract

Existence and uniqueness of strong solutions for a class of stochastic functional di fferential equations in Hilbert spaces are established. Suf cient conditions which guarantee the transference of mean square and pathwise exponential stability from stochastic partial diff erential equations to stochastic functional partial di erential equations are studied. The stability results derived are also applied to stochastic ordinary differential equations with hereditary characteristics. In particular, as a direct consequence our main results improve some of those from Mao and Shah in which it is proved that under certain conditions pathwise exponential stability is transferred from nondelay equations to delay ones if the constant time lag appearing in the problem is su ciently small, while in our treatment the transference actually holds for arbitrary bounded delay variables not only in finite but in infi nite dimensions.