Probabilistic and Analytical Research of White Noise System

描述白噪声系统的基本空间有几种构造方法，每一种方法都建立了相同的空间。这些结构并不是那么直接。组织者成功地介绍了空间的直接建造，并在国际会议上展示了结果。通过这些结果，我们可以比较Hida演算和Malliavin演算的异同。每个研究者都沿着各自领域的组织者的思想，结合白噪声系统进行研究。S.Oharu教授研究了非线性扰动半群的刻划和局部Lipshitz半群的理论。K.Taira教授研究了积分-微分算子及其对应的边值问题。特别是，他研究了它们的Feller半群和马尔可夫过程的构造。Takenaka教授研究了自相似稳定过程的结构和特征。这些谱在他的讨论中是有效的，有望在分析稳定的白噪声系统中发挥重要作用。Yamato博士研究了随机分析与无穷小分析的关系。Nakamura博士结合Hausdorff测度研究了随机分形集的Hausdorff维和Packing维。首先，我们的主要目的是阐明无限维Bargmann空间和Levy Laplace空间。但对于更基本的科目，我们得到了重要的结果。研究结果对原文主题的研究具有一定的参考价值。通过对无限维偏微分算子、半群和扩散过程的研究，我们可以对白噪声系统的结构有更深入的了解，其中一个新的发展是引入了一类新的广义函数或泛函，并与H.H.Kuo教授合作给出了特征定理。因为指数泛函是测试泛函，所以这个类足够广泛，可以用于应用程序并且功能强大。

The basic spaces to describe white noise system have been constructed in several ways, each of which sets up the same spaces. Those constructions are not so direct. The organizer succeeded to introduce a direct construction of the spaces and presented the results in international conferences. By the results we can compare the similarity and distinction between Hida calculus and Malliavin calculus.Each investigator researched along the idea of the organizer in their own field in connection with white noise system. Prof.S.Oharu investigated characterization of non-linearly perturbed semi-groups and theory of locally Lipshitz semi-groups. Prof.K.Taira studied boundary problems of integro-differential operators and corresponding to the operators. Especially, he investigated constructions of their Feller semi-groups and Markov processes. Prof.Takenaka researched constructions and characterizations of self similar stable processes. The spectra, which are effective in his discussion, are expected important roles in analysis of stable white noise system. Dr.Yamato studied stochastic analysis in the relationship with infinitesimal analysis. Dr.Nakamura researched Hausdorff dimensions and packing dimensions of random fractal sets together with Hausdorff measures.In the beginning, our main purpose was to clarify infinite dimensional Bargmann space and Levy Laplacian. But we get important results for more basic subjects. The results are useful for research of the original theme. By the investigation of infinite dimensional partial differential operators, semi-groups and diffusion processes, we can have deeper understanding of the structure of white noise systems.One of new developments is that we introduced a new class of generalized functions or functionals and gave characterization theorems by the collaboration with Prof.H.-H.Kuo. The class is sufficiently wide to use for applications and powerful because exponential functionals are test functionals.