On Limit Measures and Their Supports for Stochastic Ordinary Differential Equations

TL;DR

This paper investigates the behavior of stationary measures in stochastic ODEs, showing they concentrate on stable sets and not on unstable or saddle structures, with applications to various dynamical systems.

Contribution

It establishes conditions under which limit measures of stochastic ODEs are supported on Lyapunov stable sets, extending understanding of invariant measures in stochastic dynamics.

Findings

Limit measures avoid repellers and saddle chains.

Limit measures concentrate on Lyapunov stable invariant sets.

Applications include Morse-Smale and Axiom A systems.

Abstract

This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for SODEs to admit the Freidlin-Wentzell or Dembo-Zeitouni large deviations principle with weaker compactness condition, we prove that limit measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limit measures on acyclic saddle or trap chains. This illustrates that limit measures are concentrated on Liapunov stable compact invariant sets. Applications are made to the Morse-Smale systems, the Axiom A systems including structural stability systems and separated star systems, the gradient or gradient-like systems, those systems possessing the Poincare-Bendixson property with a finite number of limit sets to obtain that limit measures live on Liapunov stable critical elements, Liapunov stable basic sets, Liapunov stable equilibria and Liapunov stable limit sets including equilibria, limit cycles and saddle or trap cycles, respectively. A number of nontrivial examples admitting a unique limit measure are provided, which include monostable, multistable systems and those possessing infinite equivalent classes.