Nonlinear analysis on Innovation of stochastic processes and random fields

今年的研究成果涉及三个主题，总结如下。1)随机过程的创新方法。获取创新是研究进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，创新是一个附加过程的时间导数。因此，创新问题与加性过程的列维分解密切相关。因此，我们建立了理想元素随机变量系统，作为对给定随机复杂系统的创新。2)分数功率分布。人们已经认识到，分数功率分布出现在许多科学领域并发挥着重要作用。为了研究这种分布，我建议将这些分布嵌入到一个加性过程中，这样我们就可以研究产生这种分数分布的原因。在许多情况下，嵌入到一个稳定的过程是可能的。在这种情况下，可以应用稳定过程的Levy分解，其结果告诉我们分数阶功率分布现象的隐藏统计性质。3)高斯噪声和泊松噪声的对偶性。一方面，高斯白噪声和泊松噪声可以用相似的方法处理，但另一方面，它们的差异是显著的。从两种噪声的二元性可以看出不同的表现之一。关于无限对称群，我给出了这两种噪声的对偶性。

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.