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

Abstract

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 $$\begin{aligned} d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t \end{aligned}$$starting from deterministic initial conditions in \(\Omega \), under rather general assumptions on f (for instance, f may have several critical points in \(\Omega \)). This work is a continuation of the previous paper [14] where the exit point distribution from \(\Omega \) is studied when \(X_0\) is initially distributed according to the quasi-stationary distribution of \((X_t)_{t\ge 0}\) in \(\Omega \). 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.