Stochastic stability of physical measures in conservative systems

TL;DR

This paper investigates the stochastic stability of physical measures in conservative dynamical systems on manifolds, showing that physical measures are stochastically stable under small noise perturbations, with some results extending to non-physical measures.

Contribution

It establishes conditions under which invariant measures in conservative systems are stochastically stable, including cases with non-physical measures and systems on bounded domains.

Findings

Physical measures are strongly stochastically stable under small noise.

Stochastic stability can occur even for non-physical invariant measures.

Results extend to conservative systems on bounded domains.

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.