Dirichlet's and Thomson's principles for non-selfadjoint elliptic operators with application to non-reversible metastable diffusion processes

TL;DR

This paper develops variational principles for non-selfadjoint elliptic operators to estimate transition times in non-reversible diffusion processes, advancing understanding of metastability in such systems.

Contribution

It introduces two variational formulas for capacity in non-selfadjoint contexts, linking minimizers to boundary-value elliptic equations and enabling precise metastable transition estimates.

Findings

Derived sharp estimates for transition times between wells.

Linked variational minimizers to elliptic boundary-value solutions.

Enhanced understanding of metastability in non-reversible diffusions.

Abstract

We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles to provide a sharp estimate for the transition times between two different wells for non-reversible diffusion processes. This estimate permits to describe the metastable behavior of the system.