Infinite Dimensional Stochastic Processes and the Information Analysis

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We deeply appreciate the grant for scientific research (term : academic years 2003, 2004) from JSPS. In this research, we considered roles as an information analysis in researching the infinite dimensional stochastic analysis jointly from major fields probability theory, analysis, variational geometry, number theory and computer science.Main results which we obtained are the following : 1) By changing the state space, we constructed an infinite dimensional Wiener process associated with the Levy Laplacian, and moreover we can extend this Laplacian operator to an operator on a space of operator- valued white noise functionals to get 2). 2) We also constructed a quantum stochastic process generated by the quantum Levy Laplacian. Based on this result a quantum information analysis associated the Levy Laplacian can be expanded to discuss the quantum communication theory 3) By using an infinite sequence of independent Brownian motions we constructed an infinite dimensional stochastic proces … More s generated by a sum of the Levy Laplacian. 4) Introducing a compensated stochastic process as a difference of two independent Levy processes, we construct a Flock space based on the Levy Laplacian. This method can be applied to construct a quantum stochastic process. 5) The method of calculating the Feynman path integrals in quantum field theory can be formulated using white noise distribution theory, and the Levy Laplacian' and the Volterra Laplacian appear in the Schroedinger type equation. 6) We obtained a relationship between the Levy Laplacian and an infinite dimensional Fractional Ornstein-Uhlenbeck process. This relationship is important to be applied the stochastic analysis based on the Levy Laplacian for the mathematical finance. Moreover we can extend this result to get a relationship between the Laplacian and a general infinite dimensional Ornstein-Uhlenbeck process.By the above results our joint research with Professor Kuo of Louisiana State University in USA was developed research of the quantum probability theory approached by white noise operator theory. Moreover we had results on entropy of infinite sequences, which is connecting to the quantum entropy.Proceeding with the above research, we organized a seminar every week and discussed each theme in the infinite dimensional stochastic analysis and information analysis between co-researchers. We gave talks on results in this research at the international conferences in Tunisia and in Italy. Through talks and discussions in several international conferences, many researchers were interested in our results and we could start new joint works with participants in the conferences. In the international conference in Italy, I had to attend at the committee of International Association " Quantum Probability and Infinite Dimensional Analysis" as a member. We expect many developments on this research with joint researchers in near future. Less