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

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 Langevin dynamics $$d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t$$ starting from the quasi-stationary distribution in $\Omega$. In the small temperature regime ($h\to 0$) and under rather general assumptions on $f$ (in particular, $f$ may have several critical points in $\Omega$), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of $f$ on $\partial \Omega$. 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.