We consider the first exit point distribution from a bounded domain Ω of the stochastic process (X t ) t≥0 solution to the overdamped Langevin dynamicsstarting from the quasi-stationary distribution in Ω. In the small temperature regime (h → 0) and under rather general assumptions on f (in particular, f may have several critical points in Ω), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of f on ∂Ω.The proof relies on tools to study tunnelling effects in semi-classical analysis. Extensions of the results to more general initial distributions than the quasi-stationary distribution are also presented.

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