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

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 [14] 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.

我们考虑从随机过程 \((X_t)_{t\ge 0}\)解的有界域\(\Omega \)到过阻尼 Langevin 动力学的第一个出口点分布 

从\(\Omega \) 中的确定性初始条件开始 ，在对f 的相当一般的假设下（例如，f在\(\Omega \) 中可能有几个临界点）。这项工作是上一篇论文 [14] 的延续，其中研究了当\(X_0\)根据\((X_t)_{的准平稳分布进行初始分布时 \( \Omega \)的出口点分布t\ge 0}\)在 \(\Omega \)。证明基于出口点分布对初始条件的依赖性的分析结果、大偏差技术和对莫尔斯函数通用性的结果。