TY  - JOUR
ID  - Hamana2025
LB  - Hamana2025
AU  - Hamana, Yuji
TI  - Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites
ST  - Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2025
VL  - 45
IS  - 4
SP  - 471
EP  - 507
DO  - https://doi.org/10.7494/OpMath.2025.45.4.471
UR  - https://doi.org/10.7494/OpMath.2025.45.4.471
AB  - 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.
KW  - Brownian motion
KW  - hitting times and sites
KW  - asymptotic expansion
SN  - 1232-9274
ER  -