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

2 MATHERIALS - MATHematics for MatERIALS
Inria de Paris, CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
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.
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Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-03058529
Contributor : Dorian Le Peutrec <>
Submitted on : Friday, December 11, 2020 - 6:40:58 PM
Last modification on : Friday, January 15, 2021 - 5:52:47 PM

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• HAL Id : hal-03058529, version 1

### Citation

Tony Lelièvre, Dorian Le Peutrec, Boris Nectoux. The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2. 2020. ⟨hal-03058529⟩

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