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.

我们研究随机现象中的数学模型问题；随机过程、维纳路径和循环空间上的泛函、晶格自旋系统、动力系统和流体动力学模型。我们列出主要结果：1。我们在维纳空间上定义了索博列夫空间，研究了维纳路径积分和条件维纳路径积分在索博列夫空间上的正则性。得到了马尔可夫半群的比较定理，可应用于维纳空间的分析。 2．我们利用 Le Gall 提出的布朗蛇的概念，研究了有关测度值分支过程（超级过程）的结构和运算。我们研究了随机过程的创新问题，并获得了一些随机过程的例子，这些随机过程在以下意义上产生非舒适过滤：齐雷尔森.3。我们给出了流体动力学方程分岔问题的计算机辅助证明。 4．我们研究了与动力系统的分岔和混沌行为有关的向量场的退化奇异性，并在同宿轨道的分岔方面得到了新的结果。 5．利用Teichmuller空间理论研究了复杂动力系统，并通过全函数构建了复杂动力理论。 6．我们研究了格劳伯动力学在晶格自旋系统中的弛豫问题。