On Davies' conjecture and strong ratio limit properties for the heat kernel
Yehuda Pinchover
TL;DR
This paper studies strong ratio limit properties and exact long-time asymptotics of the heat kernel $k_P^{\mathcal{M}}(x,y,t)$ for a general second-order parabolic operator on a noncompact Riemannian manifold, addressing Davies' conjecture on heat-kernel ratio limits. It analyzes large-time behavior, ground-state structures, and Green-function limits through the generalized principal eigenvalue $\lambda_0$ and distinguishes symmetric (self-adjoint) and nonsymmetric cases: in the symmetric case $t \mapsto k_P^{\mathcal{M}}(x,x,t)$ is log-convex and yields $\lim_{t\to\infty} \frac{k_P^{\mathcal{M}}(x,y,t+s)}{k_P^{\mathcal{M}}(x,y,t)}=1$ for all $x,y,s$, while in the nonsymmetric case only $\liminf_{t\to\infty} \frac{k_P^{\mathcal{M}}(x,y,t+s)}{k_P^{\mathcal{M}}(x,y,t)} \le 1 \le \limsup_{t\to\infty} \frac{k_P^{\mathcal{M}}(x,y,t+s)}{k_P^{\mathcal{M}}(x,y,t)}$ hold. The work establishes equivalent criteria for the existence of ratio limits and, under the additional hypothesis (eqoned), proves Conjecture conjD. It further connects the large-time quotients to the parabolic Martin boundary and to parabolic/elliptic boundaries via the parabolic Harnack inequality (URHI), yielding boundary representations, minimal Martin functions, and potential factorization results, and discusses open problems concerning ground states and minimality.
Abstract
We study strong ratio limit properties and the exact long time asymptotics of the heat kernel of a general second-order parabolic operator which is defined on a noncompact Riemannian manifold.