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Probabilistic Methods in Geometry and Analysis

几何与分析中的概率方法

基本信息

批准号：
1712427
负责人：
Maria Gordina
金额：
$ 21万
依托单位：
University of Connecticut
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-05-01 至 2021-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712427&HistoricalAwards=false
关键词：
Probabilistic Methods Geometry Analysis

项目摘要

Stochastic processes such as Brownian motion appear in many fields of science and applications. They are used to model complex systems in biology, financial markets, and social networks. In many cases the system has natural constraints which can be described as an underlying geometric structure. This project is devoted to a study of such stochastic processes, and how their properties reflect their geometric or structural environment. Sometimes the space on which the stochastic processes live is assumed to be not just curved, but infinite-dimensional, which is a natural setting for many physical or large data environments. This project includes several directions combining probability, geometry, analysis and representation theory. One of the directions of research is to study Cameron-Martin type quasi-invariance in elliptic and subelliptic settings, and its applications to functional inequalities, smoothness of probability laws for subelliptic and singular diffusions, and unitary representations of infinite-dimensional groups such as path groups. Another direction is applying coupling techniques to hypoelliptic stochastic processes, including gradient estimates and connections with geometric and analytic techniques for hypoelliptic diffusions. Some of the proposed research is motivated by physics, especially the quantum field theory (QFT), as infinite-dimensional spaces such as loop groups and path spaces appear in the QFT.

像布朗运动这样的随机过程出现在科学和应用的许多领域。它们被用来对生物学、金融市场和社会网络中的复杂系统进行建模。在许多情况下，系统具有可描述为基本几何结构的自然约束。这个项目致力于研究这种随机过程，以及它们的性质如何反映它们的几何或结构环境。有时，随机过程所在的空间被假设为不仅是曲线的，而且是无限维的，这对于许多物理或大型数据环境来说是一个自然的设置。这个项目包括几个方向，结合概率，几何，分析和表现理论。其中一个研究方向是研究椭圆和次椭圆环境下的Cameron-Martin类拟不变性及其在泛函不等式、次椭圆和奇异扩散概率律的光滑性以及无限维群的么正表示(如路群)中的应用。另一个方向是将耦合技术应用于亚椭圆随机过程，包括梯度估计以及与亚椭圆扩散的几何和分析技术的联系。一些拟议的研究是由物理学，特别是量子场论(QFT)推动的，因为无限维空间，如环群和路径空间，出现在QFT中。