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We consider the first exit point distribution from a bounded domain $\Omega$of the stochastic process $(X_t)_{t\ge 0}$ solution to the overdamped Langevindynamics $$d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t$$ starting from thequasi-stationary distribution in $\Omega$. In the small temperature regime($h\to 0$) and under rather general assumptions on $f$ (in particular, $f$ mayhave several critical points in $\Omega$), it is proven that the support of thedistribution of the first exit point concentrates on some points realizing theminimum of $f$ on $\partial \Omega$. The proof relies on tools to studytunnelling effects in semi-classical analysis. Extensions of the results tomore general initial distributions than the quasi-stationary distribution arealso presented.

The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points