DMS Analysis and Stochastic Analysis Seminar (SASA)

Speaker: Hongyi Chen  (University of Illinois, Chicago)

Title: Comparison Geometry, Short Time Heat Kernel Asymptotics, and the Multiplicative Stochastic Heat Equation

Abstract:  We show that a regularity condition for existence and uniqueness of the Multiplicative Stochastic Heat Equation with measure valued initial conditions derived in previous work on non-positively curved compact Riemannian manifolds is in fact sharp for all compact Riemannian manifolds. In contrast to the previous scenario, estimating the Brownian Bridge density accurately requires nontrivial geometric work for both when the endpoints are far away or close to each other. In the case of near endpoints, comparison geometry is necessary to recover Euclidean (or sphere-like) behavior. At far away points, the possibility of conjugate points requires one to use different heat kernel estimates.

Based on ongoing joint work with Robert Neel (Lehigh) and Cheng Ouyang (UIC).