Davies' method for anomalous diffusions

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.