2019/06/11 by D. I. Borisov, Oskar A. Sultanov, Borisov, D. +1
Mathematics · Physics and Astronomy · #35B25 #35C20 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1906.04715
openalex publication_date 2019/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the exit time from a bounded multi-dimensional domain Ω of the stochastic process Yε=Yε(t,a), t\geqslant 0, a∈ A, governed by the overdamped Langevin dynamics dYε =-∇ V(Yε) dt +√(2)ε dW, Yε(0,a)≡ x∈Ω where ε is a small positive parameter, A is a sample space, W is a n-dimensional Wiener process. The exit time corresponds to the first hitting of ∂Ω by the trajectories of the above dynamical system and the expectation value of this exit time solves the boundary value problem (-ε2Δ+∇ V⋅ ∇)uε=1\quadin Ω, uε=0\quadon ∂Ω. We assume that 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 as well. Our main result is a complete asymptotic expansion for uε as well as for the lowest eigenvalue of the considered problem and for the associated eigenfunction. The asymptotics for uε involves a term exponentially large ε; we find this term in a closed form. Apart of this term, we also construct a power in ε asymptotic expansion such that this expansion and a mentioned exponentially large term approximate uε up to arbitrarily power of ε. We also discuss some probabilistic aspects of our results.