Phanuel Mariano

Research interests

My research lies at the intersection of probability, geometry and analysis. In particular, I am interested in using probabilistic tools to solve problems in analysis in the setting of curved spaces with degeneracies. The operators I study are hypoelliptic and their natural setting are spaces called sub-Riemannian manifolds. A probabilistic tool used in my research involves the coupling of diffusion processes. Coupling is a way of constructing Markov processes with prescribed laws on the same space. I have also been interested in applying probabilistic techniques in proving inequalities for the expected lifetime of diffusion processes and the first Dirichlet eigenvalue of a domain. One can think of the first Dirichlet eigenvalue as the fundamental frequency of a drum. These inequalities often show the beautiful connection between probability, analysis and physics.