Reducing exit-times of diffusions with repulsive interactions

TL;DR

This paper establishes a Kramers' law for the exit-times of certain self-interacting diffusions at low temperatures, revealing that non-convex interactions can reduce exit-times compared to non-interacting cases.

Contribution

It extends Kramers' law to non-convex interaction cases, showing that interactions can decrease metastable exit-times, and introduces a coupling technique for analysis.

Findings

Exit-times can be smaller with non-convex interactions.

Kramers' law applies to a broader class of diffusions.

Coupling with linear processes simplifies analysis.

Abstract

In this work we prove a Kramers' type law for the low-temperature behavior of the exit-times from a metastable state for a class of self-interacting nonlinear diffusion processes. Contrary to previous works, the interaction is not assumed to be convex, which means that this result covers cases where the exit-time for the interacting process is smaller than the exit-time for the associated non-interacting process. The technique of the proof is based on the fact that, under an appropriate contraction condition, the interacting process is conveniently coupled with a non-interacting (linear) Markov process where the interacting law is replaced by a constant Dirac mass at the fixed point of the deterministic zero-temperature process.