Stability results for symmetric jump processes on metric measure spaces with atoms. Long version

TL;DR

This paper extends stability results for symmetric jump processes and heat kernel estimates to include metric measure spaces with atoms, such as graphs, by constructing an auxiliary space that manages atomic measures.

Contribution

It generalizes existing stability results to spaces with atoms, including graphs, using an auxiliary space construction to transfer properties.

Findings

Stability of heat kernel estimates on spaces with atoms.

Construction of an auxiliary space to handle atomic measures.

Transfer of properties between original and auxiliary spaces.

Abstract

Consider a symmetric Markovian jump process $\{X_t\}$ on a metric measure space $(M, d, \mu)$. Chen, Kumagai, and Wang recently showed that two-sided heat kernel estimates and the parabolic Harnack inequality are both stable under bounded perturbations of the jumping measure, assuming $(M, d, \mu)$ satisfies the volume-doubling and reverse-volume-doubling conditions. These results do not apply if $(M, d, \mu)$ is a graph (or more generally, if $M$ contains any atoms $x$ such that $\mu(x)>0$) because it is impossible for reverse-volume-doubling to hold on a space with atoms. We generalize the results of Chen, Kumagai, and Wang to a larger class of metric measure spaces, including all infinite graphs with volume-doubling. Our main tool is the construction of an "auxiliary space" that smooths out the atoms. We show that many properties transfer from $(M, d, \mu)$ to the auxiliary space, and vice versa, including heat kernel estimates, the parabolic Harnack inequality, and their stable characterizations.