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 [25], 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 [32], which considered the same problem for metastable reversible diffusion processes. Our proof is based on the recently developed partial differential equation (PDE) approach to metastability. To the best of our knowledge, this study is the first to propose a robust methodology for applying the PDE approach to non-reversible models.Comment: 43 pages, 4 figure