Energy inequalities for cutoff functions and some applications

By S. Andres, Martin T. Barlow. 2012

67 citations recorded by the source corpus.

We consider a metric measure space with a local regular Dirichlet form. We establish necessary and sufficient conditions for upper heat kernel bounds with sub-diffusive space-time exponent to hold. This characterization is stable under rough isometries, that is it is preserved under bounded perturbations of the Dirichlet form. Further, we give a criterion for stochastic completeness in terms of a Sobolev inequality for cutoff functions. As an example we show that this criterion applies to an anomalous diffusion on a geodesically incomplete fractal space, where the well-established criterion in terms of volume growth fails.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a4f0a75d13ec8cd516697ffbd25ff36f3a65588e

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces (2013, methodology): “The only hope would be to exploit the existence of specific cut-off functions related to the exponent m as in [1].”
Two-sided estimates of heat kernels of jump type Dirichlet forms (2018, methodology): “The proof below follows essentially the argument of Andres and Barlow [1] that was done in the setting of local Dirichlet forms.”