Sharp Riemannian heat kernel estimates on the cut locus and the Parabolic Anderson model

Using sharp global heat kernel bounds and geodesic comparison geometry, we show that the Dalang condition for well-posedness of the parabolic Anderson model with measure-valued initial conditions, first introduced on Euclidean space, holds on general compact Riemannian manifolds. We furthermore establish upper and lower moment bounds for all such solutions, providing evidence for intermittency in this generality. This extends and simplifies earlier work that required non-positive curvature.