2019 Fiscal Year Final Research Report
The study of global properties of Wiener sausages for diffusion processes
| Project/Area Number |
16K05208
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Kumamoto University |
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Principal Investigator |
Hamana Yuji 熊本大学, 大学院先端科学研究部(理), 教授 (00243923)
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | 確率解析 / Bessel 過程 / Ornstein-Uhlenbeck 過程 / Wiener sausage / 到達時刻 / Bessel 関数 / 合流型超幾何関数 |
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| Outline of Final Research Achievements |
We gave the second term of the distribution of the first hitting time of Bessel process and showed analyticity of zeros of Bessel functions of the second kind with respect to the indices. In addition, we succeeded in representing the denstiy of the first hitting time of the radial part of a Brownian motion with constant drift, which is one of the generalizations of Bessel processes, in the case when the starting point is the origin.
On the other hand, the distribution function of the first hitting time of the radial part of a Ornstein-Uhlenbeck process, which is another generalization of Bessel processes, can be represented by confluent hypergeometric functions and their zeros with respect to the first parameter.
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| Free Research Field |
確率論
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| Academic Significance and Societal Importance of the Research Achievements |
Brown 運動に対する 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，球から流出した熱の総量を表す．これはある種理想的な状態で考察することになり，熱の伝わり方にひずみがあるときには，Brown 運動にドリフトをつけた確率過程に対する Wiener sausage の体積の期待値を調べれることになる．本研究では「一定の方向に熱が伝わりやすい」「中心から離れるにしたがって熱が伝わりにくい」といった状態の場合を考察して，前者には完全な解答を与えることに成功した．後者については前段階の結果を得ることができた．
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