Degenerate Diffusions and Related Heat Kernel Estimates

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This research project lies at the intersection of several areas of mathematics: probability, analysis and differential geometry. The PI will work on problems related to diffusion processes and their heat kernels in degenerate geometric settings, merging tools of the three mentioned areas. Some of the problems are pertinent to various applied fields, including mathematical finance, control of dynamic systems, machine learning, and others. The PI will integrate her research expertise into teaching probability courses, mentoring students, and organizing conferences. The PI will also continue her effort in encouraging women and other underrepresented groups in math. The project focuses on random diffusion processes that are subject to a priori non-holonomic constraints which can perfectly fit into the framework of sub-Riemannian geometry. A recurring theme of proposed topics is the interaction between the underlying geometric structure and the limiting behaviors of the associated Markov processes. The project is primarily concerned with three topics. The first is to study stochastic processes and the explicit heat kernel on sub-Riemannian model spaces, which provides great examples and strong intuition for the understanding of general cases. The second topic concerns the limiting behaviors -in both small and large time scale -of a general degenerate diffusion process, which reflect the underlying geometric information of geodesic and Ricci curvature bound respectively. The third topic is to study the heat content of bounded domains in a sub-Riemannian manifold, which reveals the geometry of the domain such as surface area and total mean curvature of the boundary.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.