Deterministic dynamical system which has an asymptotical stable equilibrium is
considered under persistent perturbation by white noise. If the perturbation does not vanish in the equilibrium position than there is not Lyapunov's stability. New concept of stability on a large time interval is discussed. The measure of stability is expectation of the square distance from the trajectory till the equilibrium position. The method of parabolic equation is applied to both estimate the expectation and prove such stability.
The results are applied to analyse the autoresonance phenomenon.