Probabilistic and Analytical Research of White Noise System

描述白色噪声系统的基本空间可以用几种方法构造，每种方法都建立相同的空间。这些结构并不那么直接。组织者成功地引入了空间的直接建设，并在国际会议上展示了成果。通过结果我们可以比较希达演算和Malliavin演算之间的相似性和区别。每个研究者都沿着组织者的想法在他们自己的领域中研究与白色噪声系统有关的问题。S.Oharu教授研究了非线性扰动半群的特征和局部Lipshitz半群的理论。平良光教授研究积分微分算子的边值问题及相应的算子。特别是，他研究了他们的费勒半群和马尔可夫过程的建设。竹中教授研究了自相似稳定过程的构造和特征。这些谱在稳定白色噪声系统的分析中具有重要作用。大和博士研究了随机分析与无穷小分析的关系。中村博士研究了随机分形集的Hausdorff维数和packing维数以及Hausdorff测度，我们最初的主要目的是阐明无限维Bargmann空间和Levy Laplacian。但我们在更基础的科目上得到了重要的结果。研究结果对原题的研究有一定的参考价值。通过对无穷维偏微分算子、半群和扩散过程的研究，我们可以对白色噪声系统的结构有更深入的了解，其中一个新的进展是我们引入了一类新的广义函数或泛函，并与H. H.Kuo.这个类足够广泛，可以用于应用程序，并且功能强大，因为指数泛函是测试泛函。

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.