Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property

Existence and uniqueness of strong solutions for a class of stochastic functional differential equations in Hilbert spaces are established. Sufficient conditions which guarantee the transference of mean-square and pathwise exponential stability from stochastic partial differential equations to stochastic functional partial differential 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 by Mao & Shah in which it was proved that under certain conditions pathwise exponential stability is transferred from non-delay equations to delay equations if the constant time-lag appearing in the problem is sufficiently small, while in our treatment the transference actually holds for arbitrary bounded delay variables not only in finite but in infinite dimensions.