Stability of delay evolution equations with fading stochastic perturbations

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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 then an exponentially stable deterministic system remains to be exponentially stable (in mean square). This idea has already been checked for ordinary linear stochastic differential equations and linear stochastic difference equations. Here a similar statement is proven for a nonlinear stochastic evolution equation with quickly enough fading stochastic perturbations. More exactly: if the level of the stochastic perturbation is given by a continuous and square integrable function, then the zero solution of the considered exponentially stable deterministic system remains exponentially mean square stable independently on the maximum magnitude of the stochastic perturbations. Applications of the obtained results to stochastic reaction-diffusion equations and stochastic 2D Navier-Stokes model are shown. Consideration of other types of fading stochastic perturbations (for instance, if stochastic perturbations fade on the infinity not so quickly) is an open problem.