Extremes of stochastic processes and random fields: new directions

这个建议的目的是研究非常一般的无限可分随机场的“极端重复集的形状”。这是一类非常广泛的模型，包含高斯，α稳定，泊松，伽马和许多其他重要的随机场作为特殊情况。“形状”的概念包括连续集的连通分量的数量，这些分量有多少个“洞”，以及什么样的，等等。无限可分随机场提供了一类很好的模型，因为它们丰富的数学结构，因为它们的连续集的可能的“形状”的丰富性。研究随机集的思想和工具来自于概率论、代数拓扑学、遍历理论、几何学和数据挖掘。该提案旨在通过使用来自不同数学领域的新颖思想来进一步我们对极端的理解。了解极端现象和相关风险 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。金融风险和安全风险也是如此。这个项目的另一个影响将是在医学上：它将为医生提供新的统计工具，通过研究在没有疾病存在的情况下，在纯粹随机的情况下，极端的形状往往会发现扫描中偏离正常的情况。

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.