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Consider a symmetric Markovian jump process $\{X_t\}$ on a metric measurespace $(M, d, \mu)$. Chen, Kumagai, and Wang recently showed that two-sidedheat kernel estimates and the parabolic Harnack inequality are both stableunder bounded perturbations of the jumping measure, assuming $(M, d, \mu)$satisfies the volume-doubling and reverse-volume-doubling conditions. Theseresults 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 forreverse-volume-doubling to hold on a space with atoms. We generalize theresults of Chen, Kumagai, and Wang to a larger class of metric measure spaces,including all infinite graphs with volume-doubling. Our main tool is theconstruction of an "auxiliary space" that smooths out the atoms. We show thatmany properties transfer from $(M, d, \mu)$ to the auxiliary space, and viceversa, including heat kernel estimates, the parabolic Harnack inequality, andtheir stable characterizations.

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