Energy inequalities for cutoff functions and some applications

TL;DR

This paper characterizes conditions for heat kernel bounds and stochastic completeness in metric measure spaces with Dirichlet forms, applying to anomalous diffusions on fractals and demonstrating stability under perturbations.

Contribution

It provides necessary and sufficient conditions for heat kernel bounds and introduces a new criterion for stochastic completeness applicable to fractal spaces.

Findings

Heat kernel bounds characterized by space-time exponents.

Stability of conditions under rough isometries.

New criterion for stochastic completeness on fractals.

Abstract

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.