Infinite Dimensional Analysis and its Applications

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We have researched the infinite dimensional stochastic analysis based on the white noise on the theme 'Infinite Dimensional Analysis and its Applications' from April 1997 to March 1998. We organized several symposiums and seminars, and discussed with co-researchers each other. In this research, we obtained fundamental progress on infinite dimensional Laplacians, in particular the Levy Laplacian. The self-adjoint operator is very important in Quantum theory. So far the Levy Laplacian has been considered as non self-adjoint operator. But in this research we succeeded in extending the Levy Laplacian to a self-adjoint operator densely defined on some Hilbert space in the space of generalized white noise functionals. Based on this result we expect to get many applications to describe phenomena. As applications we have already obtained a relation between the Levy Laplacian and the number operator and also a relation between a semi-group generated by the Laplacian and an infinite dimensional Ornstein-Uhlenbeck process. Moreover if we consider the Levy Laplacian acting on Poisson noise functionals, then more fruitful results can be obtained and those results are closely related to the mathematical finance. Those are also developments on Hida calculus and can be applied to Matsuzawa's researches on partial differential equations in terms of the infnite dimensional method. An analogue to the p-adic white noise analysis can be easily obtained by those results. Poisson noise analysis is useful to find many examples in Nishi's researches. The Levy Laplacian can be characterized by the ergodic property. This is closely related to Mimachi's researches on the ergodic theory. The self-adjointness of the Levy Laplacian made joint projects between co-researchers.