Mathematical Study on Stochastic Models

随机模型的数学研究

基本信息

批准号：
62460007
负责人：
WATANABE Shinzo
金额：
$ 3.39万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (B)
财政年份：
1987
资助国家：
日本
起止时间：
1987 至 1988
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-62460007/
关键词：
Stochastic analysis Malliavin calculus Generalized Wiener functionals Asymptotic problems of heat kernels Lie superalgebra Periodic solutions of Duffing equations Fractals 双曲的変換の離散部分群熱方程式の基本解の漸近理論 Duffing方程式周期解双曲的変換離散部分群 Wien

项目摘要

An analysis on the Wiener space (Malliavin calculus) has been studied as an analogy of Schwartz distribution theory. In this framework, the pull-back of Schwartz distributions under a non-degenerate Wiener map can be defined as generalized Wiener functionals and by using this, the regularity and the asymptotics with respect to parameters of the law of Winer functionals can be discussed. This method has been applied to obtain asymptotic results for heat kernels (the fundamental solutions of heat equations). Furthermore, by taking a finite measure space as the parameter space, we can discuss the case of geat kernels with boundary conditions. As an application, a probabilistic proof was obtained for the Gauss-Bonnet-Chern therorem in the case of manifolds with boundaries. Also, this method has been applied to asymptotic problems of degenerate heat kernels.The method of Poisson point processes with values in function spaces has been applied effectively to the study of diffusion processes with boundary conditions. By using this method in the construction problem of diffusions with Wentzell's boundary conditions, we could show the existence and uniqueness of diffusions in certain cases which could not be obtained before by other methods.A new general method has been established for the construction of irreducible unitary representations of Lie superalgebras.New results have been obtained for the existence of periodic solutions of Duffing equations. Also, the theory of interval dynamical systems and self-similar sets has been applied to obtain interesting results for the classical nowhere differentiable functions of Weierstrass and Besicovitch. This kind of research on fractals will be an important contact point of probability theory and analysis in future.Interesting examples have been obtained for discrete groups of hyperbolic motions.

已经研究了对维纳空间（Malliavin conculus）的分析，以类比为施瓦茨分布理论。在此框架中，可以将Schwartz分布在非脱位Wiener地图下的下拉可以定义为广义Wiener函数，并且通过使用此功能，可以讨论有关酿酒师函数定律参数的规则性和渐近性。已应用此方法来获得热核（热方程的基本解决方案）的渐近结果。此外，通过将有限的测量空间作为参数空间，我们可以讨论具有边界条件的GEAT内核的情况。作为一种应用，在具有边界的歧管的情况下，获得了高斯 - 骨网卫星Therorem的概率证明。同样，该方法已应用于退化热核的渐近问题。具有功能空间值的泊松点过程的方法已有效地应用于具有边界条件的扩散过程。通过在与温泽尔边界条件的扩散问题中使用这种方法，我们可以在某些情况下显示出扩散的存在和独特性，在某些情况下无法通过其他方法获得。为了存在DUFFERE方程的周期性解决方案，已获得了Like Superalgebras的表示。同样，已经应用了间隔动力学系统和自相似集的理论，以获得Weierstrass和Besicovitch的经典无处可区分功能的有趣结果。对分形的这种研究将是未来概率理论和分析的重要接触点。对于离散的双曲动作组，已经获得了感兴趣的示例。