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

Boris Nectoux (1)

1 Institute for Analysis and Scientific Computing [Wien]

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.

Mots clés: overdamped Langevin process; mean exit time; metastability; Eyring-Kramers formulas

Domaines: Mathématiques [math]; Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Boris Nectoux. Mean exit time for the overdamped Langevin process: the case with critical points on the boundary. 2019. ⟨hal-02196801v1⟩