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 $$d{X}_t=-\mathrm{\nabla }f(X_t)dt+\sqrt{2\text{ve}}d{B}_t$$ when $\text{ve}\to 0$ and in the case when $D$ contains a unique non degenerate minimum of $f$ and ${\text{pa}}_{\text{mbf}n}f>0$ on $\text{pa}D$. 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 $\text{ve}\to 0$, a sharp asymptotic estimate of the smallest eigenvalue of the operator $$L_{\text{ve}}=-\text{ve}\mathrm{\Delta }+\mathrm{\nabla }f\cdot \mathrm{\nabla }$$ associated with Dirichlet boundary conditions on $\text{pa}D$. The approach does not require $f{\mathrm{\mid }}_{\mathrm{\partial }D}$ 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}.