权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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.

布朗运动等随机过程出现在许多科学和应用领域。它们被用于模拟生物学、金融市场和社交网络中的复杂系统。在许多情况下，系统具有自然约束，这些约束可以被描述为底层几何结构。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。有时候，随机过程所处的空间被假设为不仅是弯曲的，而且是无限维的，这是许多物理或大型数据环境的自然设置。该项目包括结合概率，几何，分析和表示理论的几个方向。其中一个研究方向是研究椭圆和亚椭圆情形下的Cameron-Martin型拟不变性，及其在泛函不等式、亚椭圆和奇异扩散概率律的光滑性以及无限维群（如路群）的酉表示中的应用。另一个方向是将耦合技术应用于亚椭圆随机过程，包括梯度估计和亚椭圆扩散的几何和分析技术的连接。一些研究的动机是物理学，特别是量子场论（QFT），因为无限维空间，如循环群和路径空间出现在QFT中。