Geometry, Topology, and the Parabolic Anderson Model
Hongyi Chen, University of Illinois-Chicago
Abstract: The Parabolic Anderson Model is a stochastic PDE subject to much active research. One new direction is seeing the effects of geometry on interesting properties of the solution. I will present two works on which the equation is posed over Riemannian manifolds. In the first, the equation is posed over a compact manifold and starts from a measure. The main insight is that geodesics being finite is needed for well-posedness results similar to those already known for Euclidean space. In the second, the equation is posed over a Hadamard manifold and starts from a bounded function. Here, it is revealed that having a negative curvature upper bound uniformly bounds the moments of the solution if the noise is weak. Based on ongoing work with Fabrice Baudoin and Cheng Ouyang.
