Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues

Abstract

We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering 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. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by the eigenvalues of the classical capacity matrix of the array of capacitors made of balls of radius $ϵ$ centered at the positions of the local minima of $F$. We also get very precise uniform control on the corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision with the inverse mean metastable exit times from each minimum. In [BEGK3] it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical Eyring–Kramers formula.