Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites
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\sig...
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AGH Univeristy of Science and Technology Press
2025-07-01
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| Series: | Opuscula Mathematica |
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| Online Access: | https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4522.pdf |
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| author | Yuji Hamana |
| author_facet | Yuji Hamana |
| author_sort | Yuji Hamana |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-27159db2f25a4b02b65f20320a1ff67e |
| institution | DOAJ |
| issn | 1232-9274 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj-art-27159db2f25a4b02b65f20320a1ff67e2025-08-20T02:40:29ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742025-07-01454471507https://doi.org/10.7494/OpMath.2025.45.4.4714522Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sitesYuji Hamana0https://orcid.org/0000-0002-9997-3114University of Tsukuba, Department of Mathematics, 1-1-1 Tennodai, Tsukuba 305-8571, JapanWe 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.https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4522.pdfbrownian motionhitting times and sitesasymptotic expansion |
| spellingShingle | Yuji Hamana Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites Opuscula Mathematica brownian motion hitting times and sites asymptotic expansion |
| title | Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites |
| title_full | Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites |
| title_fullStr | Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites |
| title_full_unstemmed | Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites |
| title_short | Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites |
| title_sort | tail probability of the hitting time of brownian motion to a sphere with fixed hitting sites |
| topic | brownian motion hitting times and sites asymptotic expansion |
| url | https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4522.pdf |
| work_keys_str_mv | AT yujihamana tailprobabilityofthehittingtimeofbrownianmotiontoaspherewithfixedhittingsites |