Mean exit time for the overdamped Langevin process: the case with critical points on the boundary.

Let (X t) t ≥ 0 be the overdamped Langevin process on R d , i.e. the solution of the stochastic differential equation d X t = − ∇ f (X t) d t + h d B t. Let Ω ⊂ R d be a bounded domain. In this work, when X 0 = x ∈ Ω , we derive new sharp asymptotic equivalents (with optimal error terms) in the limit h → 0 of the mean exit time from Ω of the process (X t) t ≥ 0 (which is the solution of (− h 2 Δ + ∇ f · ∇) w = 1 in Ω and w = 0 on ∂ Ω ), when the function f : Ω ¯ → R has critical points on ∂ Ω. Such a setting is the one considered in many cases in molecular dynamics simulations. This problem has been extensively studied in the literature but such a setting has never been treated. The proof, mainly based on techniques from partial differential equations, uses recent spectral results from [Le Peutrec and Nectoux, Anal. PDE, 2020] and its starting point is a formula from the potential theory. We also provide new sharp leveling results on the mean exit time from Ω. [ABSTRACT FROM AUTHOR]