Abstract

Let ${(X_t)}_{t\ge 0}$ be the overdamped Langevin process on ${\mathbb{R}}^d,$ i.e. the solution of the stochastic differential equation $$d{X}_t=-\nabla f(X_t) dt+\sqrt{h} d{B}_t.$$ 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 Ω of the process ${(X_t)}_{t\ge 0}$ (which is the solution of $(-\frac{h}{2}\Delta +\nabla f·\nabla )w=1\mathrm{ }\text{in}\mathrm{ }\Omega \mathrm{ }\text{and}\mathrm{ }w=0\mathrm{ }\text{on}\mathrm{ }\partial \Omega$), 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 [Le Peutrec and Nectoux, Anal. PDE, 2020] and its starting point is a formula from the potential theory. We also provide new sharp leveling results on the mean exit time from Ω.

摘要

让 ${(X_{吨})}_{吨\ge 0}$ 是过阻尼朗之万过程 ${\mathbb{电阻}}^d,$ 即随机微分方程的解 $$d{X}_{吨}=-\nabla F(X_{吨}) d吨+\sqrt{H} d{乙}_{吨}.$$ 让 $\Omega \subset {\mathbb{电阻}}^d$成为有界域。在这项工作中，当$X_0=X\in \Omega ,$ 我们在极限中推导出新的锐渐近等价物（具有最佳误差项） $H\to 0$ 进程 Ω 的平均退出时间 ${(X_{吨})}_{吨\ge 0}$ （这是解决方案 $(-\frac{H}{2}\Delta +\nabla F·\nabla )瓦=1\mathrm{ }\text{在}\mathrm{ }\Omega \mathrm{ }\text{和}\mathrm{ }瓦=0\mathrm{ }\text{在}\mathrm{ }\partial \Omega$)，当函数 $F：\overline{\Omega }\to \mathbb{电阻}$ 有关键点 $\partial \Omega .$这种设置是在分子动力学模拟中的许多情况下考虑的设置。这个问题已在文献中进行了广泛研究，但从未处理过这种情况。该证明主要基于偏微分方程的技术，使用 [Le Peutrec and Nectoux, Anal.] 的最新光谱结果。PDE, 2020] 其起点是来自势理论的公式。我们还提供了关于从 Ω 的平均退出时间的新的锐利水平结果。