Limiting Behavior of Randomly Perturbed Diffusions with Invariant Repelling Surfaces

TL;DR

This paper analyzes how small random perturbations affect diffusion processes with invariant repelling surfaces, revealing metastable behaviors and invariant measure asymptotics across different time scales.

Contribution

It introduces a detailed asymptotic analysis of invariant measure densities and metastable distributions for perturbed diffusions with invariant hypersurfaces, highlighting scale-dependent behaviors.

Findings

Invariant measure densities near surfaces are characterized asymptotically.

Metastable distributions are linear combinations of ergodic measures.

Coefficients of combinations depend on time scale, not on perturbation size.

Abstract

We study small perturbations of diffusion processes in $\mathbb{R}^d$ that leave invariant a finite collection of hypersurfaces. Each surface is assumed to be repelling for the unperturbed process, and the unperturbed motion on each of the surfaces is assumed to be ergodic. These surfaces separate the space into a finite number of domains, each of which carries an invariant measure of the unperturbed process. We describe the asymptotics of the densities of the invariant measures near the invariant surfaces. We then describe the asymptotic behavior of the perturbed process: at different time scales (depending on the size of the perturbation), metastable distributions are described in terms of linear combinations of the ergodic invariant measures of the unperturbed system. The coefficients in the linear combination depend on the time scale but are shown not to depend on the perturbation.