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

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}.