Exit times densities of Bessel process

TL;DR

This paper analyzes the density functions of the first exit times of Bessel processes from specific intervals, providing explicit formulas, estimates, and asymptotics, with applications to Brownian motion in multiple dimensions.

Contribution

It introduces explicit representations and asymptotic estimates for exit time densities of Bessel processes, including applications to n-dimensional Brownian motion.

Findings

Derived explicit formulas for exit time densities.

Provided precise estimates and asymptotics.

Extended results to Brownian motion in higher dimensions.

Abstract

We examine the density functions of the first exit times of the Bessel process from the intervals [0,1) and (0,1). First, we express them by means of the transition density function of the killed process. Using that relationship we provide precise estimates and asymptotics of the exit time densities. In particular, the results hold for the first exit time of the n-dimensional Brownian motion from a ball.