Hitting times of Bessel processes

2010/09/17 by Byczkowski, Tomasz, Malecki, Jacek, Ryznar, Michal
#60J60 #60J65 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1009.3513

Abstract

Let T₁^(μ) 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^(μ)(t) of T₁^(μ), obtained in Byczkowski, Ryznar (Studia Math., 173(1):19-38, 2006), we prove sharp estimates of the density of T₁^(μ) which exibit the dependence both on time and space variables. Our result provides optimal 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.

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