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

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 specific compact Riemannian manifolds is in fact sharp for all compact Riemannian manifolds. The key observation needed to improve the argument in the previous cases is that different heat kernel upper bounds are needed when estimating the Brownian Bridge density accurately when the endpoints are close or far away from each other, and there is comparison geometry at work when analyzing the case of near endpoints. Based on ongoing joint work with Robert Neel (Lehigh) and Cheng Ouyang (UIC).


Bio
Hongyi Chen’s research sits at the intersection of stochastic analysis, SPDEs, and geometry, with a focus on the parabolic Anderson model and how curvature/global geometric features influence well-posedness and intermittency (moment growth). He completed his Ph.D. in 2025 and will be joining Aarhus University as a postdoctoral associate.
https://sites.google.com/uic.edu/hongyi-chen/about