Sharp estimate of the mean exit time of a bounded domain in the zero white noise limit

TL;DR

This paper provides a rigorous, sharp asymptotic estimate for the mean exit time of overdamped Langevin dynamics from a bounded domain as noise vanishes, extending previous formal results and analyzing the smallest eigenvalue of the associated operator.

Contribution

It offers the first rigorous proof of a known asymptotic formula for mean exit time in the zero-noise limit for reversible Langevin dynamics, without requiring Morse conditions on the boundary.

Findings

Sharp asymptotic formula for mean exit time as noise goes to zero.

Asymptotic estimate of the smallest eigenvalue of the generator operator.

Proof applicable without Morse function assumptions on the boundary.

Abstract

We prove a sharp asymptotic formula for the mean exit time from a bounded domain $D\subset \mathbb R^d$ for the overdamped Langevin dynamics $$d X_t = -\nabla f(X_t) d t + \sqrt{2\ve} \ d B_t$$ when $\ve \to 0$ and in the case when $D$ contains a unique non degenerate minimum of $f$ and $\pa_{\mbf n}f>0$ on $\pa D$. This formula was actually first derived in~\cite{matkowsky-schuss-77} using formal computations and we thus provide, in the reversible case, the first proof of it. As a direct consequence, we obtain when $\ve \to 0$, a sharp asymptotic estimate of the smallest eigenvalue of the operator $$L_{\ve}=-\ve \Delta +\nabla f\cdot \nabla$$ associated with Dirichlet boundary conditions on $\pa D$. The approach does not require $f|_{\partial D}$ to be a Morse function. The proof is based on results from~\cite{Day2,Day4} and a formula for the mean exit time from $D$ introduced in~\cite{BEGK, BGK}.