The exit from a metastable state: Concentration of the exit point distribution on the low energy saddle points, part 1

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 dynamics dX_t=−∇f(X_t)dt+hdB_t starting 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. Some estimates on the relative likelihood of these points are provided. The proof relies on tools from semi-classical analysis.