Stability of equilibrium with respect to a white noise

A system of ordinary differential equations with a local asymptotically stable equilibrium is considered. The problem of stability with respect to a persistent perturbation of the white noise type is discussed. The stability with given estimates is proved on a large time interval with a length of the order of the squared reciprocal magnitude of the perturbation. The proof is based on the construction of a barrier function for the Kolmogorov parabolic equation associated with the perturbed dynamical system.