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Non-reversible metastable diffusions with Gibbs invariant measure I: Eyring–Kramers formula

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Abstract

In this article, we prove the Eyring–Kramers formula for non-reversible metastable diffusion processes that have a Gibbs invariant measure. Our result indicates that non-reversible processes exhibit faster metastable transitions between neighborhoods of local minima, compared to the reversible process considered in Bovier et al. (J Eur Math Soc 6:399–424, 2004). Therefore, by adding non-reversibility to the model, we can indeed accelerate the metastable transition. Our proof is based on the potential theoretic approach to metastability through accurate estimation of the capacity between metastable valleys. We carry out this estimation by developing a novel method to compute the sharp asymptotics of the capacity without relying on variational principles such as the Dirichlet principle or the Thomson principle.

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Notes

  1. All the critical points of U are non-degenerate (i.e., the Hessian at each critical point is invertible) and isolated from others.

  2. The case \(\Sigma _{0}=\emptyset \) may occur, for instance, if we take \(H=H_{0}\) in Fig. 2. We can deal with this situation using our result by modifying H; see Remark 3.6(4).

  3. Indeed, our result with \(\varvec{\ell }=\varvec{0}\) strictly contains what has been established in [5]. See Remark 3.6-(3).

  4. Since \(\partial _{+}{\mathcal {B}}_{\epsilon }^{\varvec{\sigma }}\subset {\mathcal {K}}\) where \({\mathcal {K}}\) is defined in (8.6) we can bound \(\varvec{\ell }\) by the \(L^{\infty }({\mathcal {K}})\) norm of \(\varvec{\ell }.\) This argument will be used repeatedly in the remainder of the article without further mention.

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Acknowledgements

IS was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Nos. 2016K2A9A2A13003815, 2017R1A5A1015626 and 2018R1C1B6006896). JL was supported by the NRF grant funded by the Korea government (Nos. 2017R1A5A1015626 and 2018R1C1B6006896).

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

    Jungkyoung Lee & Insuk Seo

  2. Research Institute of Mathematics, Seoul National University, Seoul, Republic of Korea

    Insuk Seo

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  1. Jungkyoung Lee
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Lee, J., Seo, I. Non-reversible metastable diffusions with Gibbs invariant measure I: Eyring–Kramers formula. Probab. Theory Relat. Fields 182, 849–903 (2022). https://doi.org/10.1007/s00440-021-01102-z

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