Stochastic processes in sub-Riemannian geometry

亚黎曼几何中的随机过程

• 批准号: 2246817
• 负责人: Jing Wang
• 金额: $ 21万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246817&HistoricalAwards=false
• 关键词: Stochastic; processes; sub; Riemannian; geometry

This project lies at the intersection of the mathematical areas of probability, analysis and differential geometry. It primarily aims to investigate the geometric influence of the underlying space on the behavior of stochastic processes, particularly in non-smooth geometric settings, including sub-Riemannian geometry. It integrates novel and traditional techniques from probability and analysis. Some problems addressed in this project have relevance to control systems and data science. This project offers collaboration opportunities, as well as mentorship and training for graduate students.The research project primarily focuses on three topics. The first topic involves the exploration of geometrically meaningful stochastic functionals on Grassmannian manifolds and flag manifolds. Additionally, this topic aims to establish connections between these functionals and the degenerate diffusion processes associated with the Berard-Bergery fibration. The second research direction centers around novel aspects of rigidity theorems within sub-Riemannian geometry. In particular, the project aims to investigate a fresh analytical characterization of sub-Riemannian model spaces using heat kernels. Both compact and non-compact cases will be explored in this investigation. The third topic includes both probabilistic and analytic investigation of isoperimetric type problems for domains in metric measure spaces, considering the interplay between the spectrum of a domain and the exit time of a diffusion process from that domain.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目位于概率、分析和微分几何的数学领域的交叉点。它的主要目的是研究基础空间对随机过程行为的几何影响，特别是在非光滑几何环境下，包括次黎曼几何。它从概率和分析两方面融合了新的和传统的技术。本项目涉及的一些问题与控制系统和数据科学有关。该项目为研究生提供合作机会以及指导和培训。研究项目主要集中在三个主题。第一个主题涉及在Grassman流形和FLAG流形上的几何有意义随机泛函的探索。此外，本主题旨在建立这些泛函和与Berard-Bergery纤维相关的简并扩散过程之间的联系。第二个研究方向围绕次黎曼几何中刚性定理的新方面展开。特别是，该项目旨在研究使用热核的次黎曼模型空间的一种新的分析刻画。本调查将探讨紧凑型和非紧凑型两种情况。第三个主题包括对度量空间中域的等周型问题的概率和分析研究，考虑域的频谱和该域扩散过程的退出时间之间的相互作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。