Opuscula Math. 45, no. 4 (2025), 471-507
https://doi.org/10.7494/OpMath.2025.45.4.471
Opuscula Mathematica
Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites
Abstract. We consider \(d\)-dimensional Brownian motion \(\{B_\mu(t)\}_{t\geqq0}\) with a drift \(\mu\in\mathbb{R}^d\) and the first hitting time \(\sigma_{r,\mu}^{(d)}\) to the sphere with radius \(r\) centered at the origin. This article deals with asymptotic behavior of the probability that both \(t\lt\sigma_{r,\mu}^{(d)}\lt\infty\) and \(B_\mu(\sigma_{r,\mu}^{(d)})\in A\) occur simultaneously, and we obtain that this probability admits an asymptotic expansion in powers of \(1/t\) if \(d\geqq3\) and in that of \(1/\log t\) if \(d=2\) for large \(t\). Moreover, we investigate the case of Brownian motion with no drift.
Keywords: Brownian motion, hitting times and sites, asymptotic expansion.
Mathematics Subject Classification: 60J65, 60G40, 41A60.
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- Yuji Hamana
https://orcid.org/0000-0002-9997-3114- University of Tsukuba, Department of Mathematics, 1-1-1 Tennodai, Tsukuba 305-8571, Japan
- Communicated by Palle E.T. Jorgensen.
- Received: 2025-03-24.
- Revised: 2025-06-04.
- Accepted: 2025-06-04.
- Published online: 2025-07-16.
