Exit Times for Brownian Motion and Location Detection

Abstract

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.