Small deviation and geometric quantification

偏差小、几何量化

• 批准号: 0405855
• 负责人: Fuchang Gao
• 金额: $ 8.46万
• 依托单位: Regents of the University of Idaho
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405855&HistoricalAwards=false
• 关键词: Small; deviation; geometric; quantification

0405855Gao The PI will study small deviation probabilities for Gaussian processes. In particular, he will look at three specific problems. First, he will study the relationship between small deviations under different norms. This can allow results obtained under one norm to be modified so that they are true under a different norm. Second, he will look at the small deviation probabilities for Gaussian processes with specific forms of the covariance kernel. Finally, he will study the small deviation asymptotic behavior of some Gaussian random fields including the Brownian sheet. The PI believes that some combination of the Karhunen-Loeve expansion, Fourier analysis and geometric methods will allow significant progress to be made on each of these problems. Stochastic processes move randomly through space and time. Simple descriptions of these processes tell where they should be located as a function of time (the mean process) and what their range should be. Probabilists have studied large deviations (events where the process is far away from where it should be) for many years. Now they are turning to small deviations (events where the range of the process is much smaller than it should be). This research focuses on small deviations for Gaussian processes (processes whose underlying structure is the normal distribution).

PI将研究高斯过程的小偏差概率。他将特别关注三个具体问题。首先，他将研究不同规范下的小偏差之间的关系。这可以允许在一个范数下获得的结果被修改，使得它们在不同的范数下为真。其次，他将研究具有特定形式的协方差核的高斯过程的小偏差概率。最后，他将研究包括布朗单在内的一些高斯随机场的小偏差渐近行为。PI认为，Karhunen-Loeve展开，傅立叶分析和几何方法的某种组合将使这些问题中的每一个都取得重大进展。随机过程在空间和时间中随机移动。这些过程的简单描述告诉他们应该位于作为时间的函数（平均过程）和他们的范围应该是什么。概率学家研究大偏差（过程远离它应该在的地方的事件）已有多年。现在他们转向小偏差（过程的范围比它应该的范围小得多的事件）。本研究的重点是高斯过程（其底层结构是正态分布的过程）的小偏差。