The random case of Conley's theorem: II. The complete Lyapunov function

TL;DR

This paper extends Conley's fundamental theorem to random dynamical systems, constructing a complete Lyapunov function that characterizes the chain recurrent set and simplifies the analysis of their long-term behavior.

Contribution

It introduces a complete Lyapunov function for random dynamical systems, generalizing Conley's theorem to noncompact spaces under probabilistic assumptions.

Findings

Constructed a Lyapunov function constant on random chain recurrent sets

Proved the Lyapunov function decreases outside the chain recurrent set

Extended results to noncompact state spaces

Abstract

Conley in \cite{Con} constructed a complete Lyapunov function for a flow on compact metric space which is constant on orbits in the chain recurrent set and is strictly decreasing on orbits outside the chain recurrent set. This indicates that the dynamical complexity focuses on the chain recurrent set and the dynamical behavior outside the chain recurrent set is quite simple. In this paper, a similar result is obtained for random dynamical systems under the assumption that the base space $(\Omega,\mathcal F,\mathbb P)$ is a separable metric space endowed with a probability measure. By constructing a complete Lyapunov function, which is constant on orbits in the random chain recurrent set and is strictly decreasing on orbits outside the random chain recurrent set, the random case of Conley's fundamental theorem of dynamical systems is obtained. Furthermore, this result for random dynamical systems is generalized to noncompact state spaces.