On the Stability of Solutions of Stochastic Evolution Equations

Abstract

The main purpose of this chapter is to study the stability problem for stochastic partial differential equations (SPDE). The exponential stability in the mean square sense of the mild and strong solutions of linear SPDE was undertaken in a systematic manner by Haussman [3] and was continued by Chow [1]. Our approach is close to [1]. We provide a necessary and sufficient condition, in the linear case, for the exponential stability of the strong solution [14], [13] in the m.s. sense in terms of the existence of Lyapounov functional. Moreover, under a weak additional condition of mean square continuity of the map t → E ‖u ^ϕ(t) ‖ ² _V at t = 0 for ϕ ∈ V,, we establish the existence of the functional Λ(ϕ) with LΛ(ϕ) = −‖ϕ‖ ² _V . Here {u ^ϕ(t, t ≥ 0} is the Markov solution of a linear equation and L is the generator of the corresponding semigroup. In many interesting cases (see [9] and [10]), this continuity requirement is satisfied if the initial condition is smooth. It is also noted that the smoothness of the coefficients guarantees this condition by the works of Pardoux [13], and Rozovskii [14]. This is done in Section 2.

This author was supported in part by ONR N00014-91-J-1087.