Stochastic processes in 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.