Stabilization of Evolution Equations By Noise With Application to Partial Differential Equations of Parabolic Type.

Abstract

We consider a deterministic equation of evolution X 0 (t) = AX(t)dt, in a separable Hilbert space. We prove that if A generates a C 0 - semigroup, then this equation can be stabilized, in terms of Lyapunov exponents, by noise. Then we apply this abstract result to partial differential equations of parabolic type. We also compute the Lyapunov exponents of these PDEs, both deterministic and stochastic, as functions of the eigenvalues of the operator A. 1. Introduction. The present paper is a development of [10], where we have constructed an example of a class of partial differential equations being stabilized (in 1 Research partially supported by KBN grant 2 P03A 016 16 A.A. Kwieci'nska Stabilization of evolution equations 2 terms of Lyapunov exponents) by noise. In this paper we provide sufficient conditions for exponential stabilization of abstract evolution equations. We apply these results to parabolic partial differential equations extending thus the example from [10]. First..