Date of Thesis
Spring 2026
Description
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. The key tool driving our work is an associated Poisson Hierarchy, which establishes a connection between exit time moments of Brownian Motion and solutions to a family of PDE problems.
We show that exit time moments determine location up to symmetry for a variety of two-dimensional domains. Beginning with elliptical domains, we show that the first two exit time moments are sufficient to detect location up to symmetry. We establish similar results on parabolic domains by taking an appropriate limit of elliptical domains. Analogous results are shown for equilateral triangular domains. Finally, we show that the full sequence of exit time moments determines location on a rectangular domain.
Keywords
Brownian Motion, Exit Times, Echolocation, Laplace Operator, PDE
Access Type
Honors Thesis
Degree Type
Bachelor of Science
Major
Mathematics
Second Major
Physics
First Advisor
Jeffrey Langford
Recommended Citation
Kratz, Cole, "Exit Times for Brownian Motion and Location Detection" (2026). Honors Theses. 743.
https://digitalcommons.bucknell.edu/honors_theses/743