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.

随机过程（例如布朗运动）出现在许多科学和应用领域。它们用于建模生物学，金融市场和社交网络中的复杂系统。在许多情况下，系统具有自然限制，可以描述为一种基本的几何结构。该项目致力于研究此类随机过程，以及它们的性质如何反映其几何或结构环境。有时，假定随机过程的空间不仅是弯曲的，而且是无限维度的，这是许多物理或大数据环境的自然环境。该项目包括结合概率，几何，分析和表示理论的多个方向。研究的方向之一是在椭圆形和椭圆形设置中研究Cameron-Martin型准不变，及其在功能不平等，概率法的平稳性，概率法的平稳性以及奇异的弥漫性和奇异弥漫的平稳性，以及无限量二维基团（如路径组）的单位表征。另一个方向是将耦合技术应用于低纤维化随机过程，包括梯度估计和与几何和分析技术的连接，用于低纤维化扩散。某些提出的研究是由物理学的，尤其是量子场理论（QFT）的动机，因为无限维空间（例如环组和路径空间）出现在QFT中。