Davies' method for anomalous diffusions

2015/12/08 by Mathav Murugan, Murugan, Mathav, Laurent Saloff-Coste +1 · 1 citation
Mathematics · #60J35 #60J60 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60J35 #msc:60J60

paper · pdf · doi:10.48550/arxiv.1512.02360

13 pages, to appear in Proc AMS. Main changes in v3: Referee's comments incorporated. Introduction in friendlier to non-specialists with additional references and examples

arxiv created 2016/06/08 · arxiv updated 2016/06/09

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 \citeCKS. However, a cutoff Sobolev inequality due to Andres and Barlow \citeAB is used to bound the energy measure.

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