Random attractors and nonergodic attractors for diffusions with degeneracies

TL;DR

This paper studies the long-term behavior of diffusions with degeneracies on bounded domains, classifying possible invariant measures and limit behaviors, and introduces new techniques for analyzing transience and recurrence.

Contribution

It provides a complete classification of limiting behaviors for degenerate diffusions in low dimensions and develops a new Foster-Lyapunov method for hitting-time estimates.

Findings

Complete classification in 1D and 2D cases under hyperbolicity

Identification of conditions for multiple invariant measures

Development of new hitting-time estimates for transience and recurrence

Abstract

We consider a diffusion on a bounded domain, assuming that the system is irreducible inside the domain and that the diffusion has varying degree of degeneracy on the domain's boundary. The long-term statistical properties of typical trajectories started inside the domain may be governed by one invariant measure or more than one invariant measure. We describe various possible scenarios. In dimensions 1 and 2 under boundary hyperbolicity assumptions, we give a complete classification of the limiting behavior and answer the question whether sequential averaging involving more than one invariant distribution occurs. In all cases, we compute the set of weak limit points of empirical measures. Our hitting-time estimates used to prove transience or recurrence are based on a new version of the Foster-Lyapunov technique. Extensions to nonhyperbolic boundaries and higher dimensions are discussed and an application to growth rates in scalable networks is given.