Research on Fractional Brownian Motions by the Quantum Probabilistic Method.

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Motivated by the classical Segal-Bargmann transform in Gaussian analysis, which has been recognized as a useful way for the analysis on the Boson Fock space, we have tried to examine the classical-quantum correspondence of random variables (stochastic processes) by constructing one-mode Fock spaces and Hilbert spaces of square integrable analytic functions with respect to non-Gaussian Bargmann measures. It is essentially related to the complex moment problem for the construction of non-Gaussian Bargmann measures. Due to this reason, it would be quite difficult to develop the general way to solve our problem in a fully general form. Therefore, in this research program our consideration has been restricted to several special cases containing the Wilson's family of orthogonal polynomials under the special choice of parameters, which is in the higher hierarchy than the Meixner's family. As a result, we have succeeded to construct several non-Gaussian Bargmann measures, explicitly.