Research on Fractional Brownian Motions by the Quantum Probabilistic Method.

高斯分析中的经典西格尔-巴格曼变换被认为是分析玻色子Fock空间的一种有效方法，我们试图通过构造关于非高斯巴格曼测度的平方可积解析函数的单模Fock空间和Hilbert空间来检验随机变量（随机过程）的经典-量子对应关系。它本质上与构造非高斯巴格曼测度的复矩问题有关。由于这个原因，很难发展出一种以完全一般的形式来解决我们问题的一般方法。因此，在本研究计划中，我们的考虑仅限于在特殊参数选择下包含Wilson的正交多项式族的几个特殊情况，它比Meixner的族处于更高的层次。结果，我们成功地明确地构造了几个非高斯巴格曼测度。

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.