An Eyring-Kramers law for slowly oscillating bistable diffusions

TL;DR

This paper derives precise asymptotic formulas for the expected transition times in a two-dimensional, periodically forced bistable diffusion system with weak noise, extending Eyring-Kramers laws to non-reversible dynamics.

Contribution

It provides the first sharp Eyring-Kramers asymptotics for non-reversible, periodically forced bistable diffusions with large forcing frequencies.

Findings

Sharp asymptotics for transition times derived

Results applicable to a range of forcing frequencies

Extension of potential-theoretic methods to non-reversible systems

Abstract

We consider two-dimensional stochastic differential equations, describing the motion of a slowly and periodically forced overdamped particle in a double-well potential, subjected to weak additive noise. We give sharp asymptotics of Eyring-Kramers type for the expected transition time from one potential well to the other one. Our results cover a range of forcing frequencies that are large with respect to the maximal transition rate between potential wells of the unforced system. The main difficulty of the analysis is that the forced system is non-reversible, so that standard methods from potential theory used to obtain Eyring-Kramers laws for reversible diffusions do not apply. Instead, we use results by Landim, Mariani and Seo that extend the potential-theoretic approach to non-reversible systems.