Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence

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 (Lee and Seo in Probab Theory Relat Fields 182:849–903, 2022), 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 (Rezakhanlou and Seo in https://arxiv.org/abs/1812.02069 , 2018), which considered the same problem for metastable reversible diffusion processes. Our proof is based on the recently developed resolvent approach to metastability.