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

TL;DR

This paper extends sharp heat kernel estimates and the Dalang condition to general compact Riemannian manifolds, establishing moment bounds for the parabolic Anderson model and providing evidence for intermittency.

Contribution

It generalizes the Dalang condition and moment bounds for the parabolic Anderson model from Euclidean space to all compact Riemannian manifolds using geometric analysis.

Findings

Dalang condition holds on all compact Riemannian manifolds

Established upper and lower moment bounds for solutions

Provided evidence for intermittency in this setting

Abstract

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.