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

TL;DR

This paper analyzes the distribution of exit points from a domain for overdamped Langevin dynamics, focusing on how initial conditions and energy saddle points influence the exit behavior.

Contribution

It extends previous work by studying the exit point distribution from deterministic initial conditions under general assumptions on the potential function.

Findings

Exit points concentrate near low energy saddle points.

Results depend on initial conditions and the Morse structure of the potential.

Provides analytical and large deviation techniques for understanding exit distributions.

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 $$d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t$$ 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 \cite{DLLN-saddle1} 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.