Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times

Abstract

We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form $-ϵ\Delta +\nabla F(\cdot )\nabla$ on ${\mathbb{R}}^d$ or subsets of ${\mathbb{R}}^d$, where $F$ is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of $F$ can be related, up to multiplicative errors that tend to one as $ϵ\downarrow 0$, to the capacities of suitably constructed sets. We show that this capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of $F$ at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring–Kramers formula in dimension larger than $1$. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.