On the exit-problem for self-interacting diffusions

TL;DR

This paper analyzes the exit-time behavior of self-interacting diffusions, demonstrating uniform convergence rates of occupation measures and establishing an Arrhenius law for exit-times, extending classical Freidlin-Wentzell results.

Contribution

It provides uniform bounds on convergence rates for occupation measures and proves an Arrhenius law for exit-times in self-interacting diffusions, generalizing previous results.

Findings

Uniform convergence rate bounds independent of noise level 

An Arrhenius-type law for exit-times in self-interacting diffusions

Extension of Freidlin-Wentzell theory to self-interacting processes

Abstract

We study the exit-time from a domain of a self-interacting diffusion, where the Brownian motion is replaced by $\sigma B_t$ for a constant $\sigma$. The first part of this work consists in showing that the rate of convergence (of the occupation measure of the self-interacting process toward some explicit Gibbs measure) previously obtained in \cite{kk-ejp} for a convex confinment potential $V$ and a convex interaction potential can be bounded uniformly with respect to $\sigma$. Then, we prove an Arrhenius-type law for the first exit-time from a domain (satisfying classical hypotheses of Freidlin-Wentzell theory).