On hitting times, Bessel bridges and Schrödinger’s equation

Abstract

In this paper, we establish relationships between four important concepts: (a) hitting time problems of Brownian motion, (b) 3-dimensional Bessel bridges, (c) Schrödinger’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 Schrödinger-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.