A study of stationarity and causality based on the theory of KM_2O-Langevin equations

通过发展描述一维反射正性弱平稳过程时间演化的KMO-Langevin方程理论，我们不仅得到了涨落-耗散定理的统一数学体现，而且阐明了Alder-Wainwright效应的数学结构。在上述项目过程中，我们掌握了一种哲学——波动-耗散-原理——作为纯数学应用于应用科学的研究态度的指导原则。进一步，我们发展了多维弱平稳时间序列的km_20 - langevin方程理论。我们利用km_20 - langevin方程的理论解决了一维严格平稳时间序列的非线性预测问题，得到了一个可计算的非线性预测器算法，该算法已提交给J. Math。Soc。日本。此外，作为数据分析的应用，我们将开发一个新的项目，该项目由四部分组成：平稳分析、因果分析、熵分析和预测分析。一篇关于因果分析的作品被提交给名古屋数学。我们的下一个目标是通过上面的项目，寻找像混沌系统这样的复杂系统背后的某种动力学，然后预测它的未来。

By developing the theory of KMO-Langevin equations describing the time evolution of one-dimensional weakly stationary processes with reflection positivity, we have obtained not only a unified mathematical embodiment of the fluctuation-dissipation-theorem, but also elucidated the mathematical structure of Alder-Wainwright effect.In the course of the project above, we have grasped a philosophy-the fluctuation- dissipation-principle-as a guiding principle for the attitude of research in applying pure mathematics to applied science. Further, we have developed the theory of KM_2O-Langevin equations for the multi-dimensional weakly stationary time series. We have applied the theory of KM_2O-Langevin equations to be able to resolve the non-linear prediction problem for the one-dimensional strictly stationary time series , by obtaining a computable algorithm for the non-linear predictor, which is submitted to J. Math. Soc. Japan.Moreover, as applications to data analysis, we are going to develop a new project which consisits of the four part: the stationary analysys, the causal analysys, the entropy analysys and the prediction analysis. A work concernig causal analysis is submitted to Nagoya Math. J.Our next aim is to search certain dynamics behind complex system like chaotic system and then predict its future, by using the project above.