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Mixing of metastable diffusion processes with Gibbs invariant distribution

Jungkyoung Lee

TL;DR

This work develops a model-independent framework to analyze mixing of metastable diffusion processes with Gibbs invariant distributions in the small-noise regime. It introduces a multi-scale tree-structure that reduces the dynamics to a sequence of Markov chains ${\bf y}^{(p)}(\cdot)$ on metastable valleys, with time scales $\theta_{\epsilon}^{(p)}=\exp(d^{(p)}/\epsilon)$. The authors prove convergence in total variation distance at each scale and derive sharp asymptotics for the overall mixing time as the product of the last-scale mixing of ${\bf y}^{(\mathfrak{q})}$ and the last time scale $\theta_{\epsilon}^{(\mathfrak{q})}$, including a comparison with Eyring–Kramers formulas in the double-well case. The results provide a rigorous, quantitative description of metastable diffusion mixing, with implications for MCMC and related stochastic systems sharing Gibbs-type invariant measures.

Abstract

In this article, we study the mixing properties of metastable diffusion processes which possess a Gibbs invariant distribution. For systems with multiple stable equilibria, so-called metastable transitions between these equilibria are required for mixing since the unique invariant distribution is concentrated on these equilibria. Consequently, these systems exhibit slower mixing compared to those with a unique stable equilibrium, as analyzed in Barrera and Jara (Ann. Appl. Probab. 30:1164--1208, 2020). Our proof is based on the theory of metastability, which is a primary tool for studying systems with multiple stable equilibria. Within this framework, we compute the total variation distance between the distribution of the diffusion process and its invariant distribution for any time scale larger than $ε^{-1}$. Finally, we derive precise asymptotics for the mixing time.

Mixing of metastable diffusion processes with Gibbs invariant distribution

TL;DR

This work develops a model-independent framework to analyze mixing of metastable diffusion processes with Gibbs invariant distributions in the small-noise regime. It introduces a multi-scale tree-structure that reduces the dynamics to a sequence of Markov chains on metastable valleys, with time scales . The authors prove convergence in total variation distance at each scale and derive sharp asymptotics for the overall mixing time as the product of the last-scale mixing of and the last time scale , including a comparison with Eyring–Kramers formulas in the double-well case. The results provide a rigorous, quantitative description of metastable diffusion mixing, with implications for MCMC and related stochastic systems sharing Gibbs-type invariant measures.

Abstract

In this article, we study the mixing properties of metastable diffusion processes which possess a Gibbs invariant distribution. For systems with multiple stable equilibria, so-called metastable transitions between these equilibria are required for mixing since the unique invariant distribution is concentrated on these equilibria. Consequently, these systems exhibit slower mixing compared to those with a unique stable equilibrium, as analyzed in Barrera and Jara (Ann. Appl. Probab. 30:1164--1208, 2020). Our proof is based on the theory of metastability, which is a primary tool for studying systems with multiple stable equilibria. Within this framework, we compute the total variation distance between the distribution of the diffusion process and its invariant distribution for any time scale larger than . Finally, we derive precise asymptotics for the mixing time.
Paper Structure (38 sections, 34 theorems, 300 equations, 3 figures)

This paper contains 38 sections, 34 theorems, 300 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Metastable diffusion processes
  3. Fast mixing
  4. Slow mixing and metastability
  5. Main contribution of this article
  6. Related works
  7. Model and results
  8. Model
  9. Tree-structure
  10. Main result on metastability
  11. Main result on mixing
  12. Total variation distance
  13. Mixing time
  14. Double-well case and Eyring--Kramers formula
  15. Convergence in total variation distance
  16. ...and 23 more sections

Key Result

Theorem 2.1

Suppose that $\bm{b}=-(\nabla U+\bm{\ell})$ for some $U\in C^{3}(\mathbb{R}^{d})$ and $\bm{\ell}\in C^{2}(\mathbb{R}^{d};\,\mathbb{R}^{d})$ satisfying e_decb and e: growth. Then, we have the following.

Figures (3)

  • Figure 1.1: Example of potential $U$ with two local minima $\bm{m}_{1}$, $\bm{m}_{2}$ and a saddle point $\bm{\sigma}$.
  • Figure A.1: An example of potential function $U$
  • Figure A.2: Tree-structure associated with the potential given in Figure \ref{['fig:potential']}

Theorems & Definitions (74)

  • Theorem 2.1: LS-22
  • Definition 2.3: Tree-structure
  • Remark 2.4
  • Theorem 2.5: Convergence in total variation
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8: The lower bound $\theta_{\epsilon}^{(0)}$
  • Remark 2.9
  • Remark 2.10
  • Theorem 2.11
  • ...and 64 more