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

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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 (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.

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Notes

  1. This figure has been excerpted from [41, Figure 1.2].

  2. In this article, writing “\(a,\,b\)” implies that a and b are distinct.

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Acknowledgements

IS and JL was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Nos. 2016K2A9A2A13003815, 2017R1A5A1015626 and 2018R1C1B6006896) and the Samsung Science and Technology Foundation (Project Number SSTF-BA1901-03). The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Seoul National University, Seoul, Republic of Korea

    Jungkyoung Lee

  2. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, Republic of Korea

    Insuk Seo

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  1. Jungkyoung Lee
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Correspondence to Insuk Seo.

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Communicated by Alessandro Giuliani.

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Lee, J., Seo, I. Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence. J Stat Phys 189, 25 (2022). https://doi.org/10.1007/s10955-022-02986-4

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