Isotropic Stable Random Fields and Infinite Dimensional Stochastic Integrals

0204992 Rosinski基于无限维群表示，提出了一种分析各向同性和各向同性均匀稳定随机场的新方法.在这个项目中考虑的问题和量子物理学中的表示论，特别是麦基理论之间有许多相似之处。该项目旨在建立在这些类比，以建立各向同性稳定随机场的频谱表示，可能是最明确的形式，可用于分析，建模和计算机模拟这样的随机场。这个建议的第二部分是关于无穷维随机积分。随机过程的各种样本路径性质，如连续性和有界性，通常可以用这种无穷维随机积分的存在性来描述。建议简化和统一本方法的无穷维确定性函数的随机积分，然后将这些结果扩展到非预期的被积使用解耦技术，除其他外。这是迈克尔B教授的PI目前研究的产物。马库斯本文还提出了Hilbert空间中关于圆柱半鞅的随机积分的发展。这一发展将遵循最近的进展PI和他的合作者对radonification的圆柱半鞅。 各向同性随机场在湍流统计理论以及其他自然科学和工程领域中起着重要的作用。它们可以用来模拟三维湍流速度，速度和压力，等各向同性随机场的经典理论是不适合模拟随机现象与长程相关和高变异性，经常观察到的。因此，PI建议研究具有重尾稳定分布的各向同性随机场。在这种方法下发现了与量子物理学某些方法的令人困惑的联系;它们将被进一步研究和利用。第二部分是关于无穷维随机积分的理论和应用。这种积分的研究的重要性和必要性来自随机系统的理论，其特征在于非常大或连续的参数集。无限维的方法使这样的系统更简单，数学上容易处理。

0204992Rosinski A novel approach, based on infinite dimensional group representations, is proposed for the analysis of isotropic and isotropic homogeneous stable random fields. There are many analogies between problems considered in this project and representation theory in quantum physics, in particular, Mackey's theory. This project is intended to build on these analogies to establish spectral representation of isotropic stable random fields in possibly the most explicit form that could be used for analysis, modeling and computer simulation of such random fields. The second part of this proposal is concerned with infinite dimensional stochastic integrals. Various sample path properties of stochastic processes, such as continuity and boundedness, can often be described in terms of the existence of such infinite dimensional stochastic integrals. It is proposed to simplify and unify the present approach to infinite dimensional stochastic integrals of deterministic functions and then to extend these results to nonanticipating integrands using decoupling techniques, among others. This is an outgrowth of the present research of the PI with Professor Michael B. Marcus. Development of stochastic integrals with respect to cylindrical semimartingales in Hilbert spaces is also proposed. This development will follow a recent progress of the PI and his collaborators on the radonification of cylindrical semimartingales. Isotropic random fields play an important role in the statistical theory of turbulence and other areas of natural sciences and engineering. They can be used to model 3-dimensional turbulent velocity, velocity and pressure, etc. The classical theory of isotropic random fields is not suitable for modeling random phenomena with long range dependence and high variability, that are often observed. Therefore, the PI proposes to investigate isotropic random fields with heavy tailed stable distributions. Puzzling connections with certain methods of quantum physics are found under this approach; they will be further investigated and exploited. The second part of this proposal is concerned with the theory and applications of infinite dimensional stochastic integrals. The importance and need for the study of such integrals comes from the theory of random systems characterized by very large or continuous set of parameters. Infinite dimensional approach makes such systems simpler and mathematically tractable.