The study of the hittig time and the Wiener sausage for diffusion process

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03634
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: University of Tsukuba
• Principal Investigator: Hamana Yuji 筑波大学, 数理物質系, 教授 (00243923)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: Wiener sausage / Bessel 過程 / Ornstein-Uhlenbeck 過程 / 到達時刻 / square-root boundary / Brown 運動

Outline of Final Research Achievements

We determined the distribution function of the first hitting time of Bessel porcess to the square-root boundary and deduced its explicit form by using results on the first hitting time of the radial Ornstein-Uhlenbeck process with suitable parameters.
In addition, we gave the joint density function of the first hitting time and site of Brownian motion with and without a drift.
On the other hand, we gave the third term of the distribution function of the first hitting time of Bessel process and show that its behavior is different according the order of the Besel process. Moreover we had the asymptotic behavior of the expectation of the hitting time of hyperbolic Bessel process.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

Ornstein-Uhlenbeck 過程に対する Wiener sausage の体積の期待値は，球の内部の温度が１で外部の温度が０という初期状態で，球の内部の温度を１に保ったままのとき，中心から離れるにしたがって熱が伝わりにくい状況下での球から流出した熱の総量を表す．この期待値の研究のためには，Brown 運動の球面への到達時刻と適当な関数のその時刻までの確率積分の同時分布を調べることが重要であり，本研究は，その前段階として被積分関数が定数の場合についての結果を得ることができた．