Mathematical Sciences: Structure of Stationary Stable and Other Infinitely Divisible Processes

这个项目旨在建立和探索某些无限方差平稳过程理论和非奇异流的遍历理论之间的新联系。这种联系导致了对这种随机过程的自然分类，并为系统地研究它们提供了基础。由于基于平方可积过程调和分析的思想和技术在无穷方差情形中的应用非常有限，因此需要一种新的方法。另一方面，非奇异流遍历理论的许多结果与无穷方差平稳过程理论有着天然的对应关系。例如，著名的Hopf分解将流分解为耗散部分和保守部分，使我们能够识别和分离相应平稳过程的移动平均部分。在这些联系的基础上，我们将研究某些重要的无穷方差平稳过程的结构和路径性质。在许多随机现象的长时间行为中，当相同的行为模式以与时间无关的频率出现时，可以观察到随机平稳。例如，在有噪声的通信信道中，初始信号与随机噪声叠加，在信道的输出上给出平稳的随机过程。近年来，人们对高随机(无限方差)模型的研究越来越感兴趣。现有的调和分析方法和工具在中等随机平稳过程的研究中非常成功，但在无穷大方差情况下的适用性非常有限。本项目旨在建立和探索某些无穷方差平稳过程与非奇异流的遍历理论之间的新联系。这一新方法揭示了高度随机平稳过程的结构，有望为这类过程的研究提供一种新的方法。

This project is intended to establish and explore new connections between the theory of certain infinite variance stationary processes and the ergodic theory of nonsingular flows. This connection leads to a natural classification of such random processes and gives a base for their systematic investigation. A new approach has been necessary because the ideas and techniques based on harmonic analysis of square integrable processes have a very limited use in the infinite variance case. On the other hand, many results of the ergodic theory of nonsingular flows have their natural counterparts in the theory of infinite variance stationary processes. For example, the famous Hopf decomposition of flows into dissipative and conservative parts permits us to identify and isolate the moving average parts of the corresponding stationary processes. Building upon these connections, structural and path properties of certain important classes of infinite variance stationary processes will be investigated. Stochastic stationarity can be observed in the long-time behavior of many random phenomena when the same patterns of behavior occur with time independent frequencies. For example, in a noisy communication channel the initial signal is superimposed with a random noise giving a stationary random process on the channel's output. In recent years there has been a growing interest in the investigating of highly random (infinite variance) models. The existing methodology and tools of harmonic analysis, that have been very successful in the study of moderately random stationary processes, have shown a very limited applicability in the infinite variance case. This project is intended to establish and explore new connections between certain classes of infinite variance stationary processes and the ergodic theory of nonsingular flows. This new approach reveals the structure of highly random stationary processes and it is anticipated that it may lead to a new methodology i n the study of such processes.