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

Let (Xt) t≥0 be the overdamped Langevin process on R d , i.e. the solution of the stochastic differential equation dXt = −∇f (Xt) dt + √ h dBt. Let Ω ⊂ R d be a bounded domain. In this work, when X0 = 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 (Xt) t≥0 when the function f : Ω → R has critical points on the boundary of Ω. The proof is based on recent results from [27] and combines techniques from the potential theory and the large deviations theory. The approach also allows us to provide new sharp leveling results on the mean exit time from Ω