Small deviation and geometric quantification

PI将研究高斯过程的小偏差概率。特别是，他将研究三个具体问题。首先，他将研究不同规范下的小偏差之间的关系。这可以允许修改在一种规范下获得的结果，以便它们在不同的规范下是正确的。其次，他将研究具有特定协方差核形式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。国际和平研究所认为，卡胡宁-洛夫展开、傅立叶分析和几何方法的某种组合将使在这些问题上取得重大进展。随机过程在空间和时间中随机移动。对这些过程的简单描述告诉我们，作为时间的函数，它们应该位于哪里(平均过程)，以及它们的范围。多年来，概率论者一直在研究大偏差(过程远离应有水平的事件)。现在，他们正转向小偏差(过程的范围比应有范围小得多的事件)。这项研究集中在高斯过程(其潜在结构为正态分布的过程)的小偏差。

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).