2001 Fiscal Year Final Research Report Summary
Analysis of sample paths for stochastic processes
| Project/Area Number |
11640713
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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 | Kyoto University |
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Principal Investigator |
KUMAGAI Takashi Research Inst. for Math. Sci., Kyoto University, Associate Professor, 数理解析研究所, 助教授 (90234509)
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| Co-Investigator(Kenkyū-buntansha) |
SHIGEKAWA Ichiro Graduate School of Science, Kyoto University, Professor, 理学研究科, 教授 (00127234)
TAKAHASHI Yoichiro Research Inst. for Math. Sci., Kyoto University, Professor, 数理解析研究所, 教授 (20033889)
WATANABE Shinzo P.E. of Kyoto University, 名誉教授 (90025297)
HINO Masanori Graduate School of Informatics, Kyoto University, Lecturer, 情報学研究科, 講師 (40303888)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Stochastic processes / sample paths / large deviation / asymptotic behavior / fractal / Wiener sausage / law of the iterated logarithm / symmetric diffusion processes |
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| Research Abstract |
1. We have obtained an almost sure invariance principle for the range of random walks on 3-dimensional lattice. This result is a refinement of the known results like, central limit theorem, and various limit theorem can be deduced as a corollary to this result. For the range of random walks on 2-dimensional lattice, we have obtained the law of iterated logarithm. Instead of the expected loglog term, logloglog term appears.
This work will appear in Ann. Probab.
2. We have studied a problem that when disordered media are in a Euclidean space, how does the heat transfers from the space to the media. Applying the theory of Besov spaces, we have constructed a penetrating diffusion process which behaves like the diffusion on the media and like Brownian motion on the Euclidean space outside the media. This work appears in J. Funct. Anal. We then continue to work this problem and obtain a short time asymptotic behavior of the heat kernel for the diffusion process. We also obtain a functional type large deviation for the process.
We are now writing a paper on these results.
3. We have obtained asymptotic behavior of the transition probabilities between two sets on infinite dimensional symmetric diffusion processes. We adopted intrinsic distance instead of the usual distance between sets and our result is fairly general. In particular, we have proved the asymptotic behavior for the Orstein-Uhlenbeck processes on loop spaces on Riemannian manifolds, which had been an open problem. Our result is new also for degenerated symmetric diffusions on Euclidean spaces. This is a joint work with J. A. Ramirez and is now submitted for publication.
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