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ASYMPTOTICAL ANALYSIS FOR EXPONENTIAL FUNCTIONALS IN INFINITE DIMENSIONAL STOCHASTIC MODELS

无限维随机模型中指数泛函的渐近分析

基本信息

批准号：
16540097
负责人：
SHIGA Tokuzo
金额：
$ 2.18万
依托单位：
Tokyo Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540097/
关键词：
Parabolic Anderson model Lyapunov exponent Levy noise random environment random distribution Levy-Khinchin formula Levy-Ito representation random environment random probability distribution infinitely divisible Levy-Khinchin formula

项目摘要

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.

我们完成了“随机模型中指数泛函的渐近分析”这一研究项目，得到了如下结果：1.抛物型安德森模型的李雅普诺夫指数的渐近分析;在时空高斯白色噪声势的情况下，已经获得了相对于耦合常数的李雅普诺夫指数的渐进阶，并且在本项目中我们将其扩展到更一般的空间-对随机环境中的随机运动理论提出了一种基于随机概率分布（RPD）随机分析的新方法。这可以被认为是淬火分析和退火分析之间的中间分析。为此，我们定义了取值于概率分布集的无穷可分RPD和Levy过程，证明了无穷可分RPD的Levy-Khinchin公式，并利用一类新的Poisson积分证明了Levy过程的Levy-Ito型表示.此外，我们将该理论应用于随机环境中的随机运动，得到了新的极限定理。3.关于项目的主题，合作者得到了以下有趣的结果。(a)It证明了随机矩阵特征值点过程的标度极限与费米子点过程的标度极限一致，得到了中心极限定理和大偏差结果.（B）对于随机游动的范围，在条件概率钉扎律下得到了一个大偏差的结果。（3）粗糙路径分析得到了很大的发展，并将其应用于Ito映射的Taylor展开和无限维随机分析。