The concentration of zero-noise limits of invariant measures for stochastic dynamical systems

Abstract

We study the zero-noise concentration phenomena of invariant measures for SDE. The models are with locally Lipschitz coefficients and have more than one ergodic state. By the large deviations method and an expression of invariant measures, we estimate the invariant measures in neighborhoods of stable sets, unstable sets and their complements. Our results illustrate that the zero-noise limiting invariant measures will concentrate on the stable sets, where a cost functional W(Ki) is minimized. This implies that the long time behaviors of ODE and SDE must have different judging criteria. Furthermore, we prove the large deviations principle of invariant measures.