The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2

We consider the first exit point distribution from a bounded domain Ω of the stochastic process (Xt)t≥0 solution to the overdamped Langevin dynamics dXt=-∇f(Xt)dt+hdBtstarting from deterministic initial conditions in Ω , under rather general assumptions on f (for instance, f may have several critical points in Ω). This work is a continuation of the previous paper [14] where the exit point distribution from Ω is studied when X is initially distributed according to the quasi-stationary distribution of (Xt)t≥0 in Ω. The proofs are based on analytical results on the dependency of the exit point distribution on the initial condition, large deviation techniques and results on the genericity of Morse functions.