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We prove a sharp asymptotic formula for the mean exit time from a boundeddomain $D\subset \mathbb R^d$ 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 casewhen $D$ contains a unique non degenerate minimum of $f$ and $\pa_{\mbf n}f>0$on $\pa D$. This formula was actually first derivedin~\cite{matkowsky-schuss-77} using formal computations and we thus provide, inthe reversible case, the first proof of it. As a direct consequence, we obtainwhen $\ve \to 0$, a sharp asymptotic estimate of the smallest eigenvalue of theoperator $$L_{\ve}=-\ve \Delta +\nabla f\cdot \nabla$$ associated withDirichlet boundary conditions on $\pa D$. The approach does not require$f|_{\partial D}$ to be a Morse function. The proof is based on resultsfrom~\cite{Day2,Day4} and a formula for the mean exit time from $D$ introducedin~\cite{BEGK, BGK}.

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