Davies' method for anomalous diffusions

TL;DR

This paper adapts Davies' perturbation method to derive sub-Gaussian heat kernel bounds for anomalous diffusions, overcoming previous limitations due to singular energy measures.

Contribution

It introduces a modified Davies' method using a cutoff Sobolev inequality to handle anomalous diffusions, providing new heat kernel bounds.

Findings

Established sub-Gaussian upper bounds for anomalous diffusion heat kernels

Modified Davies' method applicable to singular energy measures

Utilized cutoff Sobolev inequality for energy measure bounds

Abstract

Davies' method of perturbed semigroups is a classical technique to obtain off-diagonal upper bounds on the heat kernel. However Davies' method does not apply to anomalous diffusions due to the singularity of energy measures. In this note, we overcome the difficulty by modifying the Davies' perturbation method to obtain sub-Gaussian upper bounds on the heat kernel. Our computations closely follow the seminal work of Carlen, Kusuoka and Stroock \cite{CKS}. However, a cutoff Sobolev inequality due to Andres and Barlow \cite{AB} is used to bound the energy measure.