2006 Fiscal Year Final Research Report Summary
ASYMPTOTICAL ANALYSIS FOR EXPONENTIAL FUNCTIONALS IN INFINITE DIMENSIONAL STOCHASTIC MODELS
| Project/Area Number |
16540097
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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 | Tokyo Institute of Technology |
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Principal Investigator |
SHIGA Tokuzo Tokyo Institute of Technology, Graduate School of Science and Technology, Professor, 大学院理工学研究科, 教授 (60025418)
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| Co-Investigator(Kenkyū-buntansha) |
HAMANA Yuji Kumamoto University, Faculty of Science, Professor, 理学部, 教授 (00243923)
SHIRAI Tomoyuki Kyushu University, Graduate School of Matematical Science, Associate Professor, 大学院数理学研究院, 助教授 (70302932)
NOMURA Yuji Tokyo Institute of Technology, Graduate School of Science and Engineering, Assistant Professor, 大学院理工学研究科, 助手 (40282818)
INAHAMA Yuzuru Tokyo Institute of Technology, Graduate School of Science and Engineering, Assistant Professor, 大学院理工学研究科, 助手 (80431998)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Parabolic Anderson model / Lyapunov exponent / Levy noise / random environment / random distribution / Levy-Khinchin formula / Levy-Ito representation |
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| Research Abstract |
We performed this research project on "Asymptotical analysis for exponential functionals in stochastic models", and obtained the following results.
1.Asymptotical analysis of the Lyapunov exponent of the Paraboloc Anderson model ; In the case of space-time Gaussian white noise potential the asymptotical order of the Lyapunov exponent relativet to the coupling constant has been obtained and in this project we extend it to more general space-time Levy noise and obtained an precise asymptotical order together with reasonable interpretation of the constant appearing in it.
2.For the theory of random motions in random environments we proposed a new approach, which is based upon a stochastic analysis of random probability distributions (RPD),. This may be regarded as an intermediate one between quenched analysis and annealed analysis. For this aim we defined infinitely divisible RPDs and Levy processes taking values in the set of probability distributions, and we proved Levy-Khinchin formula for infinitely divisible RPDs, and Levy-Ito type representation for the Levy processes using a new type of Poisson integrals. Furthermore we applied this theory to a random motion in a random environment and obtained new type of limit theorems.
3.Concerning the theme of the project collaborators obtained the following interesting results.
(a)It was proved that a scaling limit of point process of eigenvalues of random matrices is identified with the Fermion point process, for which the central limit theorem and large deviation results were obtained.
(b)Concernning of the range of random walks, a large deviation result was obtained under a conditional probability law of pinning.
(c)Rough path analysis was developed largely, and applied it to Taylor expansion of Ito maps and related to infinite-dimensional stochastic analysis.
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