Stochastic stability of measures in gradient systems

Abstract

Stochastic stability of a compact invariant set of a finite dimensional, dissipative system is studied in our recent work “Concentration and limit behaviors of stationary measures” (Huang et al., 2015) for general white noise perturbations. In particular, it is shown under some Lyapunov conditions that the global attractor of the systems is always stable under general noise perturbations and any strong local attractor in it can be stabilized by a particular family of noise perturbations. Nevertheless, not much is known about the stochastic stability of an invariant measure in such a system. In this paper, we will study the issue of stochastic stability of invariant measures with respect to a finite dimensional, dissipative gradient system with potential function f. As we will show, a special property of such a system is that it is the set of equilibria which is stable under general noise perturbations and the set Sf of global minimal points of f which is stable under additive noise perturbations. For stochastic stability of invariant measures in such a system, we will characterize two cases of f, one corresponding to the case of finite Sf and the other one corresponding to the case when Sf is of positive Lebesgue measure, such that either some combined Dirac measures or the normalized Lebesgue measure on Sf is stable under additive noise perturbations. However, we will show by constructing an example that such measure stability can fail even in the simplest situation, i.e., in 1-dimension there exists a potential function f such that Sf consists of merely two points but no invariant measure of the corresponding gradient system is stable under additive noise perturbations. Crucial roles played by multiplicative and additive noise perturbations to the measure stability of a gradient system will also be discussed. In particular, the nature of instabilities of the normalized Lebesgue measure on Sf under multiplicative noise perturbations will be exhibited by an example.