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

Abstract

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(−ε2Δ+∇V⋅∇)uε=1inΩ,uε=0on∂Ω,ε→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 uε. The asymptotics for uε involves an exponentially large term, which we find in a closed form. We also construct a power in ε asymptotic expansion such that this expansion and a mentioned exponentially large term approximate uε up to arbitrary power of ε.