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

Abstract

Let be the overdamped Langevin process on i.e. the solution of the stochastic differential equation Let be a bounded domain. In this work, when we derive new sharp asymptotic equivalents (with optimal error terms) in the limit of the mean exit time from Ω of the process (which is the solution of ), when the function has critical points on 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.