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Pré-Publication, Document De Travail Année : 2022

Mean exit time for the overdamped Langevin process: the case with critical points on the boundary

Boris Nectoux
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Résumé

Let $(X_t)_{t\ge 0}$ be the overdamped Langevin process on $\mathbb R^d$, i.e. the solution of the stochastic differential equation $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t.$$ Let $\Omega\subset \mathbb R^d$ be a bounded domain. In this work, when $X_0=x\in \Omega$, we derive new sharp asymptotic equivalents (with optimal error terms) in the limit $h\to 0$ of the mean exit time from $\Omega$ of the process $(X_t)_{t\ge 0}$ (which is the solution of $(-\frac h2 \Delta+\nabla f\cdot \nabla)w=1 \text{ in } \Omega \text{ and } w=0 \text{ on } \pa \Omega$), when the function $f:\overline \Omega\to \mathbb R$ has critical points on $\partial \Omega$. Such a setting is the one considered in many cases in molecular dynamics simulations. This problem has been extensively studied in the literature but such a setting has never been treated. The proof, mainly based on techniques from partial differential equations, uses recent spectral results from~\cite{pcbord} and its starting point is a formula from the potential theory. We also provide new sharp leveling results on the mean exit time from~$\Omega$.
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Dates et versions

hal-02196801 , version 1 (29-07-2019)
hal-02196801 , version 2 (08-10-2020)
hal-02196801 , version 3 (27-02-2021)
hal-02196801 , version 4 (07-04-2022)

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  • HAL Id : hal-02196801 , version 4

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Boris Nectoux. Mean exit time for the overdamped Langevin process: the case with critical points on the boundary. 2022. ⟨hal-02196801v4⟩
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