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

2 MATHERIALS - MATHematics for MatERIALS
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria de Paris
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.
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https://hal.archives-ouvertes.fr/hal-03728053
Contributor : Tony Lelièvre Connect in order to contact the contributor
Submitted on : Wednesday, July 20, 2022 - 5:44:53 AM
Last modification on : Tuesday, August 2, 2022 - 3:21:09 AM

### Identifiers

• HAL Id : hal-03728053, version 1
• ARXIV : 2207.09284

### Citation

Tony Lelièvre, Dorian Le Peutrec, Boris Nectoux. Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary. 2022. ⟨hal-03728053⟩

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