Hitting Times of Bessel Processes

Abstract

Let \(T_1^{(\mu)}\) be the first hitting time of the point 1 by the Bessel process with index μ ∈ ℝ starting from x > 1. Using an integral formula for the density \(q_x^{(\mu)}(t)\) of \(T_1^{(\mu)}\), obtained in Byczkowski and Ryznar (Stud Math 173(1):19–38, 2006), we prove sharp estimates of the density of \(T_1^{(\mu)}\) which exhibit the dependence both on time and space variables. Our result provides optimal uniform estimates for the density of the hitting time of the unit ball by the Brownian motion in ℝn, which improve existing bounds. Another application is to provide sharp estimates for the Poisson kernel for half-spaces for hyperbolic Brownian motion in real hyperbolic spaces.