This work is concerned with a class of semilinear stochastic functional parabolic differential equations of retarded type. We first establish conditions to ensure the existence of a unique non-negative solution of the stochastic delay partial differential equation under investigation....

The aim of this paper is to investigate exponential stability of paths for a class of Hilbert space-valued non-linear stochastic evolution equations. The analyses consist in using exponential martingale formula, Lyapunov functional and some special inequalities derived for our stability...

A strong solutions approximation approach for mild solutions of stochastic functional differential equations with Markovian switching driven by Lévy martingales in Hilbert spaces is considered. The Razumikhin-Lyapunov type function methods and comparison principles are studied in pursuit of...

Sufficient conditions for almost surely asymptotic stability with a certain decay function of sample paths, which are given by mild solutions to a class of semilinear stochastic evolution equations, are presented. The analysis is based on introducing approximating system with strong solution and...

Existence, uniqueness and continuity of mild solutions are established for stochastic linear functional differential equations in an appropriate Hilbert space which is particularly suitable for stability analysis. An attempt is made to obtain some infinite dimensional stochastic extensions of...

Some criteria for the mean square and almost sure exponential stability of nonlinear stochastic partial differential equations are shown in this paper. In particular, the main results obtained in Caraballo and Real (1994, Stochast. Anal. Appl. 12(5), 517-525) are improved, since the new...