Nonlinear analysis on Innovation of stochastic processes and random fields

本年度的研究成果主要涉及三个方面，现将其归纳如下：1）随机过程的新息方法获得新息是研究演化随机复杂系统的最佳途径之一，因此该问题的研究从去年开始，一直延续到今年。在重要的情况下，新息是一个可加过程的时间导数。因此，新息问题与可加过程的Levy分解有着密切的联系.因此，我们建立了理想化基本随机变量系统，作为对已知随机复杂系统的创新。2）分数阶幂分布分数阶幂分布在许多科学领域都有广泛的应用，并发挥着重要的作用。为了研究这个分布，我建议将这些分布嵌入到一个可加过程中，这样我们就可以研究为什么会出现这样一个分数分布。在许多情况下，嵌入到稳定的进程中是可能的。在这些情况下，可以应用稳定过程的Levy分解，其结果告诉我们具有分数幂分布的现象的隐藏的统计特性。3）高斯噪声和泊松噪声之间的对偶性高斯白色噪声和泊松噪声可以以类似的方式处理，但另一方面是显著不同的。两种噪声之间的对偶性可以看出它们的不相似性。关于无限对称群，我给出了这两个噪声的对偶。

The research result of this year involves three topics as are summarized below.1) Innovation approach to stochastic processes.To obtain innovation is one of the best ways to investigate evolutional random complex systems, so that the research of this problem started last year and has been carried over to this year. In the important cases, the innovation is the time derivative of an additive process. Innovation problem has therefore close connections with the Levy decomposition of an additive process. Thus, we have established the system of idealized elemental random variables as the innovation of the given random complex system.2) Fractional power distributions.It has been recognized that fractional power distributions appear in many fields of science and play important roles. For the study of this distribution, I have proposed to embed those distributions into an additive process, so that we can study the reason why such a fractional distribution arises. There are many cases where embedding into a stable process is possible. In those cases, the Levy decomposition of the stable process can be applied and the results tell us hidden statistical properties of the phenomena with fractional power distribution.3) Duality between Gaussian and Poisson noises.Gaussian white noise and Poisson noise can be dealt with in a similar manner on the one hand, but dissimilarity is significant on the other. One of the realizations of dissimilarity can be seen in the duality between the two noises. In connection with the infinite symmetric group, I have given a duality of the two noises.