Brownian Hitting to Spheres

Abstract

Let $S_r^{d-1}$ be the sphere in $\bR^d$ whose center is the origin and the radius is $r$, and ${\sigma }_r$ be the first hitting time to it of the standard Brownian motion $\{B_t{\}}_{t\geqq 0}$, possibly with constant drift. The aim of this article is to show explicit formulae by means of spherical harmonics for the density of the joint distribution of $({\sigma }_r,B_{{\sigma }_r})$ and to study the asymptotic behavior of the distribution function.

13 pages