Infinite Dimensional Stochastic Analysis and its Applications

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Supported by Grant-in-Aid for Scientific Research(C)we obtained fruitful results on an infinite dimensional stochastic analysis based on the Levy Laplacian. The Laplace equation associated the Levy Laplacian is equivalent to the yang-Mills equation in quantum physics. Moreover, the Laplacian is closely related to the square power of the quantum white noise. To research a stochastic analysis based on the Laplacian has an important role of describing and understanding the micro-phenomena. With our previous researches in 1999 we obtained several results on a stochastic analysis based on the Laplacian in this research : Diagnalization of the Levy Laplacian, Constructing the domain(nuclear space)on which the Laplacian has a continuous spectrum, Constructing a stochastic process and a quantum stochastic process generated by the Laplacian on the domain, Constructing a stochastic process generated by functions of the Laplacian, etc. Those stochastic processes are infinite but the parts of fluctuation are finite and therefore we expect to apply this theory to other fields in Engineering. Moreover we can describe some biological phenomena and economical phenomena as stochastic models using the Levy Laplacian. If we change elements of the domain of the Levy Laplacian to some operators, then we can research the quantum white noise theory based on the Laplacian. We hope that this direction of researches has more furuitful developments.Supported by the Grant we attended at international workshops in Italy and in USA to give lectures on recent results on infinite dimensional stochastic analysis, and also gave talks at Kyoto University, Kyushu University and others. We asked Prof.Ito at Hiroshima Women's University to give a review on applications of white noise theory to the economy.