Stochastic stability of measures in gradient systems

•Stochastic stability of invariant measures is investigated for gradient systems.•Noise vanishing limits of Gibbs measures are discussed.•Examples of non-convergence or instability are given for both additive and multiplicative noise perturbations.

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 ff. 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 SfSf of global minimal points of ff which is stable under additive noise perturbations. For stochastic stability of invariant measures in such a system, we will characterize two cases of ff, one corresponding to the case of finite SfSf and the other one corresponding to the case when SfSf is of positive Lebesgue measure, such that either some combined Dirac measures or the normalized Lebesgue measure on SfSf 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 11-dimension there exists a potential function ff such that SfSf 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 SfSf under multiplicative noise perturbations will be exhibited by an example.