Infinite Dimensional Stochastic Processes and the Information Analysis

我们非常感谢JSPS的科研资助（任期：2003、2004学年）。在本研究中，我们从概率论、分析学、变分几何、数论和计算机科学等主要领域联合起来，将角色作为一种信息分析来研究无穷维随机分析。得到的主要结果如下：1)通过改变状态空间，构造了一个与Levy拉普拉斯算子相关的无限维Wiener过程，并将该拉普拉斯算子扩展为算子值白噪声泛函空间上的算子，得到2)。2)构造了一个由量子Levy拉普拉斯算子生成的量子随机过程。在此基础上，将Levy Laplacian的量子信息分析扩展到量子通信理论；3)利用独立布朗运动的无穷序列，构造了由Levy Laplacian和生成的无限维随机过程；4)引入补偿随机过程作为两个独立Levy过程的差分，构造了基于Levy Laplacian的Flock空间。该方法可用于构造量子随机过程。5)量子场论中计算费曼路径积分的方法可以用白噪声分布理论表述，在薛定谔型方程中出现了Levy Laplacian’和Volterra Laplacian。6)得到了Levy Laplacian与无限维分数阶Ornstein-Uhlenbeck过程的关系。这一关系对于应用基于Levy Laplacian的数学金融随机分析具有重要意义。此外，我们可以推广这一结果，得到拉普拉斯过程与一般无限维Ornstein-Uhlenbeck过程之间的关系。在此基础上，我们与美国路易斯安那州立大学郭教授共同开展了基于白噪声算子理论的量子概率论研究。此外，我们还得到了无限序列熵的结果，这与量子熵有关。在上述研究的基础上，我们每周组织一次研讨会，共同探讨无限维随机分析和信息分析中的各个主题。我们在突尼斯和意大利的国际会议上就这项研究的结果发表了演讲。通过在几次国际会议上的会谈和讨论，许多研究人员对我们的研究结果感兴趣，我们可以与会议参与者开始新的合作工作。在意大利的国际会议上，我不得不作为成员参加国际协会“量子概率与无限维分析”委员会。我们期待在不久的将来与联合研究人员在这项研究上取得许多进展。少

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