Sharp estimate of the mean exit time of a bounded domain in the zero white noise limit

2017/10/20 by Boris Nectoux, Nectoux, Boris · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Diffusion and Search Dynamics #FOS: Mathematics #Markov Chains and Monte Carlo Methods #math.AP

paper · pdf · doi:10.48550/arxiv.1710.07510

17 pages

openalex publication_date 2017/10/20 · openalex created_date 2017/11/10 · arxiv created 2018/07/10 · arxiv updated 2018/07/11 · openalex updated_date 2026/07/28

Abstract

We prove a sharp asymptotic formula for the mean exit time from a bounded domain D⊂ \mathbb R^d for the overdamped Langevin dynamics d X_t = -∇ f(X_t) d t + √(2\ve) d B_t when \ve → 0 and in the case when D contains a unique non degenerate minimum of f and \pa_{\mbf n}f>0 on \pa D. This formula was actually first derived in~\citematkowsky-schuss-77 using formal computations and we thus provide, in the reversible case, the first proof of it. As a direct consequence, we obtain when \ve → 0, a sharp asymptotic estimate of the smallest eigenvalue of the operator L_\ve=-\ve Δ+∇ f⋅ ∇ associated with Dirichlet boundary conditions on \pa D. The approach does not require f|_{∂ D} to be a Morse function. The proof is based on results from~\citeDay2,Day4 and a formula for the mean exit time from D introduced in~\citeBEGK, BGK.

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• Eyring-Kramers formula for the mean exit time of non-Gibbsian elliptic processes: the non characteristic boundary case

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