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

Mingyi Hou

TL;DR

This work develops a variational capacity framework for a broad class of Fokker-Planck-Kolmogorov operators in divergence form that may be non-self-adjoint or degenerate, establishing a symmetric capacity notion and a corresponding variational representation. It then derives rough a priori bounds for the equilibrium potential using this capacity framework and proves sharp capacity asymptotics in the elliptic, non-reversible setting, culminating in a non-self-adjoint Eyring-Kramers formula for metastable transitions. The results extend prior reversible and non-reversible capacity analyses to a general divergence-form setting and provide a constructive approach for sharp asymptotics via a saddle-point linearization and careful energy estimates. These findings advance potential-theoretic methods for metastability in non-self-adjoint diffusions and have potential implications for understanding sharp transition times in stochastic gradient dynamics with momentum and related non-equilibrium systems.

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).

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

TL;DR

This work develops a variational capacity framework for a broad class of Fokker-Planck-Kolmogorov operators in divergence form that may be non-self-adjoint or degenerate, establishing a symmetric capacity notion and a corresponding variational representation. It then derives rough a priori bounds for the equilibrium potential using this capacity framework and proves sharp capacity asymptotics in the elliptic, non-reversible setting, culminating in a non-self-adjoint Eyring-Kramers formula for metastable transitions. The results extend prior reversible and non-reversible capacity analyses to a general divergence-form setting and provide a constructive approach for sharp asymptotics via a saddle-point linearization and careful energy estimates. These findings advance potential-theoretic methods for metastability in non-self-adjoint diffusions and have potential implications for understanding sharp transition times in stochastic gradient dynamics with momentum and related non-equilibrium systems.

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).

Paper Structure

This paper contains 16 sections, 17 theorems, 115 equations.

Table of Contents

  1. Introduction and Main Results
  2. Contributions and Novelty
  3. Preliminaries
  4. Main Results
  5. Variational Capacity
  6. Quantitative Capacity Estimates for the Elliptic Case
  7. Outline of the paper
  8. The Variational Capacity
  9. Motivation for the Capacity
  10. Proof of the Variational Capacity
  11. Rough Bounds on the Equilibrium Potential
  12. Sharp Capacity Estimates
  13. Linearization Near the Saddle Point and the Geometry
  14. Proof of Theorem 3
  15. Estimation of I
  16. ...and 1 more sections

Key Result

Lemma 1.1

Let assump:ab hold, and let $\bm{n}_{\Omega}$ be the unit outward normal and $\widehat{\bm{n}}_{\Omega}$ be the part corresponding to the last $n_0$ coordinates. Let $\Gamma(\Omega)$ and $\Gamma^\dag(\Omega)$ be the regular parts (where all continuous solutions attain the prescribed boundary values)

Theorems & Definitions (37)

  • Lemma 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 1.5
  • Definition 1.6: Capacity
  • Remark 1.7
  • Lemma 1.8
  • Theorem 1: Variational Capacity
  • Remark 1.9
  • ...and 27 more