Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary

TL;DR

This paper rigorously proves that the exit rates of overdamped Langevin dynamics from a basin through boundary saddle points follow Eyring-Kramers laws as the noise parameter approaches zero, validating jump Markov models in molecular simulations.

Contribution

It establishes the mathematical validity of Eyring-Kramers laws for exit rates at boundary saddle points in overdamped Langevin dynamics, including cases with boundary saddle points.

Findings

Exit rates follow Eyring-Kramers laws as noise vanishes.

Results apply to boundary saddle points on the domain boundary.

Supports the use of jump Markov models in molecular dynamics.

Abstract

Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t$$ and let $\Omega \subset \mathbb R^d $ be the basin of attraction of a local minimum of $f: \mathbb R^d \to \mathbb R$. Up to a small perturbation of $\Omega$ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $\Omega$ through each of the saddle points of $f$ on $\partial \Omega$ can be parametrized by the celebrated Eyring-Kramers laws, in the limit $h \to 0$. This result provides firm mathematical grounds to jump Markov models which are used to model the evolution of molecular systems, as well as to some numerical methods which use these underlying jump Markov models to efficiently sample metastable trajectories of the overdamped Langevin dynamics.