On hitting times, Bessel bridges, and Schrodinger's equation

TL;DR

This paper explores deep connections between Brownian motion hitting times, Bessel bridges, Schrödinger's equation, and heat equations with moving boundaries, offering new insights and solutions in stochastic processes and PDEs.

Contribution

It establishes novel relationships among these concepts using Girsanov's theorem and Feynman-Kac representation, including a fundamental solution for PDEs with linear potential.

Findings

Relates hitting times of Brownian motion to Bessel bridges via Girsanov's theorem

Derives a fundamental solution for PDEs with linear potential

Suggests a link between Generalized Airy processes and hitting times

Abstract

In this paper we establish relationships between four important concepts: (a) hitting time problems of Brownian motion, (b) 3-dimensional Bessel bridges, (c) Schr\"odinger's equation with linear potential, and (d) heat equation problems with moving boundary. We relate (a) and (b) by means of Girsanov's theorem, which suggests a strategy to extend our ideas to problems in $\mathbb{R}^n$ and general diffusions. This approach also leads to (c) because we may relate, through a Feynman-Kac representation, functionals of a Bessel bridge with two Schr\"odinger-type problems. In particular, we also find a fundamental solution to a class of parabolic partial differential equations with linear potential. Finally, the relationship between (c) and (d) suggests a possible link between Generalized Airy processes and their hitting times.