Variational Formulae for the Capacity Induced by Second-Order Elliptic Differential Operators

Abstract

We review recent progress in potential theory of second-order elliptic operators and on the metastable behavior of Markov processes.There has been many recent progress in the potential theory of non-reversible Markov processes. We review in this article some of these advances. In Section 1, we present a brief historical overview of potential theory and we introduce the main notions which will appear throughout the article. In Section 2, we present two variational formulae for the capacity between two sets induced by second-order elliptic operators non necessarily selfadjoint. In the following three sections we present applications of these results. In Section 3, we discuss recurrence of Markov processes; in Section 4, we present a sharp estimate for the transition time between two wells in a dynamical system randomly perturbed; and in Section 5, we prove the metastable behavior of this process.

Metastable Markov chains

“…Theorem 14.2 appeared in Gaudillière and Landim [67], and Theorem 14.3 is due to Slowik [118]. Similar Dirichlet and Thomson principles are available in the context of diffusions processes, [90,85]. Remark 14.5.…”

Asymptotics for scaled Kramers–Smoluchowski equations in several dimensions with general potentials

“…We would like to emphasize that the tools developed in Theorem 5.3 are also useful for analyzing metastable random processes, which are processes with multiple stable equilibria. It has been noted in [7,8,11,15] that, by investigating the inhomogeneous version of our main theorem (Theorem 5.3), one can obtain a complete analysis of the metastability of such processes. In addition, this method turns out to be extremely effective in the investigation of metastable diffusions.…”