Stochastic processes in sub-Riemannian geometry

该项目位于概率，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随机过程的行为的基础空间的几何影响，特别是在非光滑的几何设置，包括亚黎曼几何。它集成了概率和分析的新颖和传统技术。该项目中解决的一些问题与控制系统和数据科学有关。本项目为研究生提供了合作机会，以及指导和培训。研究项目主要集中在三个主题上。第一个主题涉及探索几何有意义的随机泛函上的格拉斯曼流形和旗流形。此外，本主题的目的是建立这些泛函和退化扩散过程与Berard-Bergery纤维化之间的联系。第二个研究方向是围绕亚黎曼几何中刚性定理的新方面。特别是，该项目的目的是研究一个新的分析表征亚黎曼模型空间使用热内核。紧和非紧的情况下，将探讨在这项调查。第三个主题包括概率和分析调查的等周型问题的域度量措施spaces.This奖项的频谱域和出口时间的扩散过程从该domain.This奖项之间的相互作用反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.