Probabilistic and Analytical Research of White Noise System

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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.