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Research on random phenomena by the methods of modern mathematics

用现代数学方法研究随机现象

基本信息

批准号：
07404005
负责人：
WATANABE Shinzo
金额：
$ 16.45万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07404005/
关键词：
Sobolev spaces on Wiener space measure valued branching processes Brownian snake innovation problem bifurcations for equations in fluid dynamics bifurcation problem in dynamical system complex dynamics of entirefunctions lattice spin system

项目摘要

We studied on problems of mathematical models in random phenomena ;stochastic processes, functionals on Wiener path and loop spaces, lattice spin systems, dynamical systems and models in fluid dynamics. We list main results :1. We defined Sobolev spaces on Wiener space and studied regularity of Wiener path integrals and conditional Wiener path integrals in terms of Sobolev spaces.We obtained a comparison theorem for Markovian semigroups which can be applied to analysis on Wiener space.2. We studied structures and operations concerning measure valued branching processes (superprocesses) by using the notion of Brownian snakes due to Le Gall.We studied the innovation problem for stochastic processes and obtained some example of stochastic processes which generate non-cosy filtrations in the sense of Tsirelson.3. We gave a computer aided proof in bifurcation problems for equations of fluid dynamics.4. We studied on a degenerating singularity of vector fields inconnection with bifurcations and chaotic behavior of dynamical systems.Also we obtained new results on bifurcations of homoclinic orbits.5. We studied complex dynamical systems by the theory of Teichmuller spaces and constructed a theory of complex dynamics by entire functions.6. We studied relaxation problem of Glauber dynamics in lattice spin systems.

我们研究了随机现象中的数学模型、随机过程、Wiener路径和回路空间上的泛函、格子自旋系统、动力系统和流体动力学模型。主要结果如下：1.定义了Wiener空间上的Soblev空间，研究了Wiener路径积分和条件Wiener路径积分在Sobolev空间上的正则性，得到了马氏半群的一个比较定理，该定理可用于Wiener空间上的分析。利用布朗蛇的概念研究了测值分枝过程(超过程)的结构和运算，研究了随机过程的新息问题，得到了一些产生Tsirelson意义下非coy滤子的随机过程的例子。给出了流体力学方程分歧问题的计算机辅助证明。研究了矢量场的退化奇异性与动力系统的分叉和混沌行为的关系，得到了关于同宿轨分叉的新结果。利用TeichMuller空间理论研究了复杂动力系统，建立了基于整函数的复杂动力学理论。研究了晶格自旋系统中Glauber动力学的松弛问题。