We consider the first exit point distribution from a bounded domain Ω of the stochastic process ${(X_t)}_{t\ge 0}$ solution to the overdamped Langevin dynamics$d{X}_t=-\mathrm{\nabla }f(X_t)dt+\sqrt{h}d{B}_t$ starting from the quasi-stationary distribution in Ω. In the small temperature regime ($h\to 0$) and under rather general assumptions on f (in particular, f may have several critical points in Ω), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of f on ∂Ω. Some estimates on the relative likelihood of these points are provided. The proof relies on tools from semi-classical analysis.

中文翻译：

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从亚稳态退出：低能量鞍点上的出口分布集中，第1部分

我们考虑随机过程的有界域Ω的第一个出口点分布 ${（X_{Ť}）}_{Ť\ge 0}$ 过度阻尼的Langevin动力学的解决方案$d{X}_{Ť}=-\mathrm{\nabla }F（X_{Ť}）dŤ+\sqrt{H}d{乙}_{Ť}$从Ω的准平稳分布开始。在小温度范围内（$H\to 0$），并且在关于f的相当一般的假设下（特别是f可能在Ω处有几个临界点），事实证明，对第一个出口点分布的支持集中在某些点上，实现了∂Ω上f的最小值。提供了关于这些点的相对可能性的一些估计。证明依赖于来自半经典分析的工具。