Stochastic processes in sub-Riemannian geometry

这个项目位于概率、分析和微分几何数学领域的交叉点。它的主要目的是研究底层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中，包括亚黎曼几何。它整合了从概率和分析的新颖和传统的技术。在这个项目中解决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。该研究项目主要集中在三个主题上。第一个主题是探索格拉斯曼流形和旗流形上几何意义的随机泛函。此外，本课题旨在建立这些功能与与Berard-Bergery纤颤相关的简并扩散过程之间的联系。第二个研究方向集中在亚黎曼几何中刚性定理的新方面。特别是，该项目旨在利用热核研究亚黎曼模型空间的一种新的分析表征。紧凑型和非紧凑型的情况下，将探讨在这个调查。第三个主题包括度量测量空间中域的等周型问题的概率和解析研究，考虑域的谱和从该域扩散过程的退出时间之间的相互作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.