On stochastic perturbations of dynamical systems with a "rough" symmetry. Hierarchy of Markov chains

TL;DR

This paper studies the long-term behavior of stochastic dynamical systems with rough symmetry, showing how hierarchies of Markov chains replace cycles in describing metastability and stochastic resonance effects.

Contribution

It introduces a framework for analyzing systems with rough symmetry, replacing cycle hierarchies with Markov chain hierarchies for long-time behavior analysis.

Findings

Hierarchy of Markov chains describes long-term dynamics with rough symmetry.

Metastable distributions can be characterized via invariant measures.

The approach applies to systems with a finite number of extreme points.

Abstract

We consider long-time behavior of dynamical systems perturbed by a small noise. Under certain conditions, a slow component of such a motion, which is most important for long- time evolution, can be described as a motion on the cone of invariant measures of the non- perturbed system. The case of a finite number of extreme points of the cone is considered in this paper. As is known, in the generic case, the long-time evolution can be described by a hierarchy of cycles defined by the action functional for corresponding stochastic processes. This, in particular, allows to study metastable distributions and such effects as stochastic resonance. If the system has some symmetry in the logarithmic asymptotics of transition probabilities (rough symmetry),the hierarchy of cycles should be replaced by a hierarchy of Markov chains and their invariant measures.