| Project/Area Number |
20540121
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kumamoto University |
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Principal Investigator |
HAMANA Yuji 熊本大学, 大学院・自然科学研究科, 教授 (00243923)
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| Project Period (FY) |
2008 – 2011
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| Project Status |
Completed (Fiscal Year 2011)
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| Budget Amount *help |
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2010: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2009: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | ランダムウォーク / 不規則媒質 / ブラウン運動 / Wiener sausage / ベッセル過程 / 大偏差原理 / エントロピー関数 / ベッセル関数 / ラプラス変換 / 到達時刻 / 無限分解可能分布 / レヴィ測度 / ランダム媒質 / 再帰性 / Wiener sausae / 漸近展開 / 変形ベッセル関数 / 漸近挙動 |
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| Research Abstract |
On this research we investigated the expectation of the cylindrical set determined by a Brownian motion, which is so-called Wiener sausage. This object has been investigated for long time in connection with heat conduction problems. We first deduced asymptotic behaviors of the expected volume of the Wiener sausage up to time t as t tends to zero and infinity.
In addition, we represented the probability that the first hitting time of the stochastic process generalized from the radial motion of the Brownian motion, which is called the Bessel process, is not large than t by means of the Bessel functions of the second kind and their zeros. By the resulting formula, we obtain the asymptotic behavior of the probability as t tends to infinity.
With the help of the modified method used to obtain the representation, we can derive the mean volume of the Wiener sausage up to time t by means of the Bessel functions of the second kind and their zeros. Moreover, this result is expected to derive its asymptotic expansion for large time.
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