Non-reversible Metastable Diffusions with Gibbs Invariant Measure I: Eyring-Kramers Formula

TL;DR

This paper proves the Eyring-Kramers formula for non-reversible metastable diffusions with Gibbs invariant measure, showing that non-reversibility accelerates transitions between local minima compared to reversible cases.

Contribution

It introduces a novel method to estimate capacities sharply without variational principles, advancing the understanding of metastability in non-reversible diffusions.

Findings

Non-reversible processes have faster metastable transition rates.

The new capacity estimation method avoids traditional variational principles.

Non-reversibility can be used to accelerate metastable transitions.

Abstract

In this article, we prove the Eyring-Kramers formula for non-reversible metastable diffusion processes that have a Gibbs invariant measure. Our result indicates that non-reversible processes exhibit faster metastable transitions between neighborhoods of local minima, compared to the reversible process considered in [Bovier, Eckhoff, Gayrard, and Klein, J. Eur. Math. Soc. 6: 399-424, 2004]. Therefore, by adding non-reversibility to the model, we can indeed accelerate the metastable transition. Our proof is based on the potential theoretic approach to metastability through accurate estimation of the capacity between metastable valleys. We carry out this estimation by developing a novel method to compute the sharp asymptotics of the capacity without relying on variational principles such as the Dirichlet principle or the Thomson principle.