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

## TL;DR

This paper analyzes the distribution of exit points for a stochastic process in a domain, showing that at low temperatures, the exit points concentrate on the lowest energy saddle points on the boundary.

## Contribution

It proves that the exit point distribution concentrates on minimal energy saddle points under general conditions, extending previous results to broader initial distributions.

## Key findings

- Exit points concentrate on minimal energy saddle points at low temperature.
- The support of the exit distribution localizes on points minimizing the potential on the boundary.
- Results are extended to initial distributions beyond the quasi-stationary state.

## 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 the quasi-stationary distribution in $\Omega$. In the small temperature regime ($h\to 0$) 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 $\partial \Omega$. The proof relies on tools to study tunnelling effects in semi-classical analysis. Extensions of the results to more general initial distributions than the quasi-stationary distribution are also presented.