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 $\mathrm{\Omega }$ of the stochastic process $(X_t{)}_{t\ge 0}$ solution to the overdamped Langevin dynamics $$d{X}_t=-\mathrm{\nabla }f(X_t)dt+\sqrt{h}d{B}_t$$ starting from deterministic initial conditions in $\mathrm{\Omega }$, under rather general assumptions on $f$ (for instance, $f$ may have several critical points in $\mathrm{\Omega }$). This work is a continuation of the previous paper \cite{DLLN-saddle1} where the exit point distribution from $\mathrm{\Omega }$ is studied when $X_0$ is initially distributed according to the quasi-stationary distribution of $(X_t{)}_{t\ge 0}$ in $\mathrm{\Omega }$. 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.