Asymptotic analysis of mean exit time for dynamical systems with a single well potential

We study the mean exit time from a bounded multi-dimensional domain Ω of the stochastic process governed by the overdamped Langevin dynamics. This mean exit time solves the boundary value problem ( − e 2 Δ + ∇ V ⋅ ∇ ) u e = 1 in Ω , u e = 0 on ∂ Ω , e → 0 . The function V is smooth enough and has the only minimum at the origin contained in Ω; the minimum can be degenerate. At other points of Ω, the gradient of V is non-zero and the normal derivative of V at the boundary ∂Ω does not vanish. Our main result is a complete asymptotic expansion for u e . The asymptotics for u e involves an exponentially large term, which we find in a closed form. We also construct a power in e asymptotic expansion such that this expansion and a mentioned exponentially large term approximate u e up to arbitrary power of e.