布朗运动的首出时间与位置检测

中文摘要 AI 辅助

2023年，Wyman和Xi提出了“你能在流形上听到自己的位置吗？”的问题。我们提出了一个相关问题：是否可以利用概率数据来恢复流形内的位置？更确切地说，我们询问是否可以从布朗运动样本路径的首出时间分布中恢复位置。我们证明，对于多种二维区域，首出时间矩可将位置确定至对称性。从边界为圆锥曲线的（凸）区域开始，我们证明在椭圆、抛物线和双曲线区域中，前两个首出时间矩足以将位置检测至对称性。我们在无界楔形区域和等边三角形区域中建立了类似结果。我们证明，在矩形区域中，完整的首出时间矩序列可确定位置。我们以关于一般平面区域的两个位置检测结果作为结尾。推动我们工作的关键工具是“泊松层级”，它建立了布朗运动的首出时间矩与一类偏微分方程（PDE）问题解之间的联系。

英文摘要

In 2023, Wyman and Xi asked ``Can you hear your location on a manifold?'' We pose a related question, asking if probabilistic data can be used to recover location within a manifold. More precisely, we ask if location can be recovered from the distribution of exit times of Brownian sample paths. We show that exit time moments determine location up to symmetry for a variety of two-dimensional domains. Beginning with (convex) domains whose boundaries are conic curves, we show that in elliptic, parabolic, and hyperbolic domains, the first two exit time moments are sufficient to detect location up to symmetry. We establish similar results in unbounded wedge domains and equilateral triangular domains. We show that the full sequence of exit time moments determines location in rectangular domains. We end with two location detection results on general planar domains. The key tool driving our work is a ``Poisson hierarchy,'' which establishes a connection between exit time moments of Brownian motion and solutions to a family of PDE problems.

发表机构

• Bucknell University（巴克内尔大学）

机构由 AI 辅助整理，请以论文原文为准。