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 Omega of the stochastic process (X-t)(t >= 0), solution to the overdamped Langevin dynamics dX(t) = -del f(X-t)dt + root h dB(t) starting from the quasi-stationary distribution in Omega. In the small temperature regime (h -> 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 derivative Omega. Some estimates on the relative likelihood of these points are provided. The proof relies on tools from semi-classical analysis. (C) 2019 Elsevier Masson SAS. All rights reserved.