Stochastic stability of physical measures in conservative systems

Abstract

Given the significance of physical measures in understanding the complexity of dynamical systems as well as the noisy nature of real-world systems, investigating the stability of physical measures under noise perturbations is undoubtedly a fundamental issue in both theory and practice. The present paper is devoted to the stochastic stability of physical measures for conservative systems on a smooth, connected, and closed Riemannian manifold. It is assumed that a conservative system admits an invariant measure with a positive and mildly regular density. Our findings affirm, in particular, that such an invariant measure has strong stochastic stability whenever it is physical, that is, for a large class of small random perturbations, the density of this invariant measure is the zero-noise limit in $L^{1}$ of the densities of unique stationary measures of corresponding randomly perturbed systems. Stochastic stability in a stronger sense is obtained under additional assumptions. Examples are constructed to demonstrate that stochastic stability could occur even if the invariant measure is non-physical. The high non-triviality of constructing such examples asserts the sharpness of the stochastic stability conclusion. Similar results are established for conservative systems on bounded domains.