Translator Disclaimer
November 2020 On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates
Naotaka Kajino, Mathav Murugan
Ann. Probab. 48(6): 2920-2951 (November 2020). DOI: 10.1214/20-AOP1440

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 (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. 338 (2003) 11–40). We also prove that in the contrary case of walk dimension two, that is, where full off-diagonal 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.

Citation

Download Citation

Naotaka Kajino. Mathav Murugan. "On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates." Ann. Probab. 48 (6) 2920 - 2951, November 2020. https://doi.org/10.1214/20-AOP1440

Information

Received: 1 October 2019; Revised: 1 March 2020; Published: November 2020
First available in Project Euclid: 20 October 2020

MathSciNet: MR4164457
Digital Object Identifier: 10.1214/20-AOP1440

Subjects:
Primary: 31E05, 35K08, 60G30
Secondary: 28A80, 31C25, 60J60

Rights: Copyright © 2020 Institute of Mathematical Statistics

JOURNAL ARTICLE
32 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.48 • No. 6 • November 2020
Back to Top