2011 Fiscal Year Final Research Report
Asymptotic behavior of random walks in random environment
| 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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| 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 |
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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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