On Schr\"odinger's equation, 3-dimensional Bessel bridges, and passage time problems

TL;DR

This paper explores the relationship between the first-passage time density of a Wiener process with a moving boundary, three-dimensional Bessel bridges, and solutions to the heat equation, offering bounds for these quantities.

Contribution

It establishes a novel connection between first-passage time densities, Bessel bridges, and heat equation solutions with moving boundaries.

Findings

Derived bounds for passage time densities.

Linked Bessel bridges to heat equation solutions.

Enhanced understanding of stochastic processes with moving boundaries.

Abstract

In this work we relate the density of the first-passage time of a Wiener process to a moving boundary with the three dimensional Bessel bridge process and a solution of the heat equation with a moving boundary. We provide bounds.