Stability of delay evolution equations with fading stochastic perturbations

Abstract

Stability of nonlinear delay evolution equation with stochastic perturbations is considered. It is shown that if the level of stochastic perturbations fades on the infinity, for instance, if it is given by square integrable function, then an exponentially stable deterministic system remains to be exponentially stable (in mean square). Applications of the obtained results to stochastic reaction-diffusion equations and stochastic 2D Navier-Stokes model are shown.