Moment estimates for the stochastic heat equation on Cartan-Hadamard manifolds

TL;DR

This paper investigates how curvature influences the stochastic heat equation on Cartan-Hadamard manifolds, establishing bounds on solution moments based on heat kernel properties and curvature conditions.

Contribution

It introduces a family of space-colored, time-white noises to analyze regularity and derives moment bounds linked to curvature and heat kernel estimates.

Findings

Exponential upper bounds for solution moments over time.

Lower bounds for moments based on sectional curvature.

Noise strength requirements depend on curvature conditions.

Abstract

We study the effect of curvature on the Parabolic Anderson model by posing it over a Cartan-Hadamard manifold. We first construct a family of noises white in time and colored in space parameterized by a regularity parameter $\alpha$, which we use to explore regularity requirements for well-posedness. Then, we show that conditions on the heat kernel imply an exponential in time upper bound for the moments of the solution, and a lower bound for sectional curvature imply a corresponding lower bound. These results hold if the noise is strong enough, where the needed strength of the noise is affected by sectional curvature.