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Davies' method for anomalous diffusions


Authors: Mathav Murugan and Laurent Saloff-Coste
Journal: Proc. Amer. Math. Soc. 145 (2017), 1793-1804
MSC (2010): Primary 60J60, 60J35
DOI: https://doi.org/10.1090/proc/13324
Published electronically: October 18, 2016
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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 (1987). However, a cutoff Sobolev inequality due to Andres and Barlow (2015) is used to bound the energy measure.


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Additional Information

Mathav Murugan
Affiliation: Department of Mathematics, University of British Columbia and Pacific Institute for the Mathematical Sciences, Vancouver, British Columbia V6T 1Z2, Canada
Email: mathav@math.ubc.ca

Laurent Saloff-Coste
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Email: lsc@math.cornell.edu

DOI: https://doi.org/10.1090/proc/13324
Received by editor(s): February 3, 2016
Received by editor(s) in revised form: May 26, 2016, and June 6, 2016
Published electronically: October 18, 2016
Additional Notes: Both authors were partially supported by NSF grant DMS 1404435
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2016 American Mathematical Society