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arXiv · 2301.03756

Brownian Hitting to Spheres

Yuji Hamana·Hiroyuki Matsumoto
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Abstract

Let $S^{d-1}_r$ be the sphere in $\bR^d$ whose center is the origin and the radius is $r$, and $σ_r$ be the first hitting time to it of the standard Brownian motion $\{B_t\}_{t\geqq0}$, 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 $(σ_r,B_{σ_r})$ and to study the asymptotic behavior of the distribution function.

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Yuji Hamana, Hiroyuki Matsumoto. 2023-01-10. Brownian Hitting to Spheres. https://arxiv.org/abs/2301.03756

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