[Paper Review] Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence

Jungkyoung Lee, Insuk Seo|arXiv (Cornell University)|Aug 19, 2020

Advanced Mathematical Modeling in Engineering15 references4 citations

TL;DR

This paper establishes the convergence of a non-reversible metastable diffusion process with Gibbs invariant measure to a continuous-time Markov chain on metastable valleys, using a resolvent approach. The key result is the rigorous derivation of the limiting generator and transition rates, extending reversible metastability theory to non-reversible dynamics governed by a Morse potential with solenoidal drift.

ABSTRACT

This article considers a class of metastable non-reversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper [32], we proved the Eyring-Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably time-rescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of [45], which considered the same problem for metastable reversible diffusion processes. Our proof is based on the recently developed resolvent approach to metastability.

Motivation & Objective

• To extend the metastable behavior analysis of reversible diffusions to non-reversible processes with Gibbs invariant measures.
• To address the lack of analytical tools for non-reversible metastable dynamics, which are known to mix faster but lack rigorous scaling limits.
• To establish a time-rescaled limit where the non-reversible diffusion converges to a continuous-time Markov chain on the deepest metastable valleys.
• To generalize previous results on reversible processes (e.g., [45]) to non-reversible settings using the resolvent approach.
• To derive explicit asymptotic expressions for the limiting transition rates between metastable states under non-reversibility.

Proposed method

• Uses the resolvent approach to metastability, a recent framework for analyzing rare transitions in diffusion processes.
• Analyzes the generator of the diffusion process and constructs a limiting generator via spectral analysis in the small noise limit.
• Applies the Eyring–Kramers formula (previously derived in [32]) as a foundation for transition time asymptotics.
• Performs asymptotic expansions of integrals over transition interfaces using Laplace's method and Gaussian approximations.
• Introduces a change of variables and decomposition of phase space into metastable valleys and transition regions.
• Employs a perturbative analysis of the drift field and Hessian structure to compute the limiting transition rates between valleys.

Experimental results

Research questions

• RQ1 Can the metastable behavior of non-reversible diffusions with Gibbs invariant measures be described by a limiting Markov chain?
• RQ2 How do the transition rates between metastable valleys in non-reversible dynamics differ from those in the reversible case?
• RQ3 What is the precise asymptotic form of the limiting generator for non-reversible metastable diffusions?
• RQ4 Can the resolvent approach be extended to non-reversible processes with solenoidal drifts satisfying ∇U·ℓ ≡ 0 and ∇·ℓ ≡ 0?
• RQ5 What role does the non-reversible drift ℓ play in modifying the effective transition rates between metastable states?

Key findings

• The time-rescaled non-reversible diffusion process converges weakly to a continuous-time Markov chain on the set of deepest metastable valleys as ε → 0.
• The limiting transition rate from valley i to j is given by an explicit formula involving the Hessian of U, the drift ℓ, and the vector field v, with the leading-order term proportional to (L H⁻¹ v) · e₁.
• The transition rate is asymptotically of order ε^(−(d+1)/2) and depends on the non-reversible drift through the term (L H⁻¹ v) · e₁.
• The resolvent approach successfully handles non-reversible dynamics, overcoming the lack of detailed balance and symmetry in the generator.
• The convergence is established via asymptotic analysis of integrals over transition interfaces, with error terms controlled uniformly in ε.
• The limiting generator is non-symmetric, reflecting the non-reversibility of the original diffusion, and captures the correct metastable transition dynamics.