On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates

TL;DR

This paper proves that for certain symmetric diffusions, sub-Gaussian heat kernel estimates imply the energy measures are singular, while Gaussian estimates imply they are mutually absolutely continuous, confirming a conjecture.

Contribution

It establishes a clear link between heat kernel estimates and the singularity or absolute continuity of energy measures in symmetric diffusions.

Findings

Sub-Gaussian estimates imply energy measures are singular.

Gaussian estimates imply mutual absolute continuity.

Verifies a conjecture by M. T. Barlow.

Abstract

We show that for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger than two (\emph{sub-Gaussian} estimates) imply the singularity of the energy measures with respect to the symmetric measure, verifying a conjecture by M.\ T.\ Barlow in [Contemp.\ Math., vol.\ 338, 2003, pp.\ 11--40]. We also prove that in the contrary case of walk dimension two, i.e., where full off-diagonal \emph{Gaussian} estimates of the heat kernel hold, the symmetric measure and the energy measures are mutually absolutely continuous in the sense that a Borel subset of the state space has measure zero for the symmetric measure if and only if it has measure zero for the energy measures of all functions in the domain of the Dirichlet form.