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

这个项目的目的是建立和探索某些无穷方差平稳过程的理论和非奇异流的遍历理论之间的新联系。 这种联系导致了这种随机过程的自然分类，并为其系统研究奠定了基础。 一种新的方法是必要的，因为基于平方可积过程的调和分析的思想和技术在无穷方差的情况下使用非常有限。 另一方面，非奇异流的遍历理论的许多结果在无穷方差平稳过程理论中有其自然的对应物。 例如，著名的霍普夫分解流到耗散和保守的部分，使我们能够识别和隔离相应的平稳过程的移动平均部分。 建立在这些连接，结构和路径性质的某些重要类别的无穷方差平稳过程将进行调查。 当相同的行为模式以与时间无关的频率发生时，可以在许多随机现象的长期行为中观察到随机平稳性。 例如，在有噪声的通信信道中，初始信号与随机噪声叠加，从而在信道的输出上给出平稳随机过程。 近年来，人们对高度随机（无穷方差）模型的研究越来越感兴趣。 调和分析的现有方法和工具，已经非常成功地在中等随机平稳过程的研究中，已经显示出非常有限的适用性在无限方差的情况下。 这个项目的目的是建立和探索某些类的无穷方差平稳过程和非奇异流的遍历理论之间的新的联系。 这一新方法揭示了高度随机平稳过程的结构，可望为研究这类过程提供一种新的方法。

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.