Stability of Mild Solutions of Stochastic Evolution Equations with Variable Delay

Abstract

Abstract In this paper, we consider the existence and stability problems associated with semilinear stochastic evolution equations with variable delay in infinite dimensions. To be precise, we first study an existence result and then the exponential stability of a mild solution as well as asymptotic stability in probability of its sample paths. Such results are established employing a comparison principle under less restrictive hypothesis than the Lipschitz condition on the nonlinear terms. An application is included to illustrate the theory. Keywords: Stochastic evolution equations with variable delaySuccessive approximationsComparison principleExistence and uniqueness of a solutionAsymptotic stability in probabilityExponential stability in mean