Sharp estimate of the mean exit time of a bounded domain in the zero white noise limit
Résumé
We prove a sharp asymptotic formula for the mean exit time from an open bounded domain D ⊂ R d for the overdamped Langevin dynamics dX t = −−f (X t)dt + √ 2ε dB t in the limit ε → 0 and in the case when D contains a unique non degenerate minimum of f and ∂ n f > 0 on ∂D. As a direct consequence, one obtains in the limit ε → 0, a sharp asymptotic estimate of the smallest eigenvalue of the operator L ε = −ε∆ + f · associated with Dirichlet boundary conditions on ∂D. The approach does not require f | ∂D to be a Morse function. The proof is based on results from [7,8] and a formula for the mean exit time from D introduced in the potential theoretic approach to metastability [4, 5].
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