Extremes of stochastic processes and random fields: new directions

这个提议的目的是研究非常一般的无限可分随机场的“极端超越集的形状”。这是一个非常广泛的模型类别，包含高斯，稳定，泊松，伽玛和许多其他重要的随机场作为特殊情况。“形状”的概念包括超越集的连接组件的数量，这些组件有多少“孔”，以及什么样的“孔”，等等。无限可分随机场由于其丰富的数学结构和其超越集的可能“形状”的丰富性，提供了一类优秀的模型。研究超越集的思想和工具将来自概率论、代数拓扑、遍历理论、几何和数据挖掘。这一建议旨在通过使用来自不同数学领域的新思想来进一步加深我们对极端的理解。了解极端现象和与这些现象相关的风险对社会非常有益：分析、预测和评估像2005年卡特里娜飓风这样的事件的影响是很难高估的。金融风险、安全风险也是如此。这个项目的另一个影响将是在医学上：它将给医生提供新的统计工具，通过研究在完全随机的情况下，当没有疾病存在时，极端的形状，来检测扫描中偏离标准的情况。

This aim of this proposal is to investigate ``the shape of the extreme exceedance sets'' of very general infinitely divisible random fields. This is a very broad class of models, containing Gaussian, alpha-stable, Poisson, Gamma and many other important random fields as special cases. The notion of``shape'' includes the number of connected components of the exceedance sets, how many ``holes'' do these components have, and of what kind, etc. The infinitely divisible random fields provide an excellent class of models because of their rich mathematical structure and because of the richness of the possible ``shapes'' of their exceedance sets. The ideas and the tools that will be employed to investigate the exceedance sets will come from probability theory, algebraic topology, ergodic theory, geometry and data mining. This proposal aims to further our understanding of the extremes by using novel ideas coming from diverse areas of mathematics. Understanding extremal phenomena and the risks associated with these phenomena is highly beneficial for the society: analysis, forecasting and evaluating the impact of events like Hurricane Katrina of 2005 is hard to overestimate. The same is true for financial risks and security risks. Another impact of this project would be in medicine: it will give doctors new statistical tools to detect departures from the norm in scans by studying the shape the extremes tend to take under pure randomness, when no disease is present.