Variational Formulation and Capacity Estimates for Non-Self-Adjoint Fokker-Planck Operators in Divergence Form

TL;DR

This paper develops a variational framework for non-self-adjoint Fokker-Planck operators, enabling capacity estimates and deriving the Eyring-Kramers formula for these operators, extending classical results to a broader class.

Contribution

It introduces a variational formulation for non-self-adjoint Fokker-Planck operators and applies it to derive capacity estimates and the Eyring-Kramers formula in this general setting.

Findings

Established a variational capacity framework for non-self-adjoint operators

Provided rough estimates for equilibrium potentials in elliptic cases

Derived the Eyring-Kramers formula for non-self-adjoint elliptic operators

Abstract

We introduce a variational formulation for a general class of possibly degenerate, non-self-adjoint Fokker-Planck operators in divergence form, motivated by the work of Albritton et al. (2024), and prove that it is suitable for defining the variational capacity. Using this framework, we establish rough estimates for the equilibrium potential in the elliptic case, providing a novel approach compared to previous methods. Finally, we derive the Eyring-Kramers formula for non-self-adjoint elliptic Fokker-Planck operators in divergence form, extending the results of Landim et al. (2019) and Lee & Seo (2022).