Stability results for symmetric jump processes on metric measure spaces with atoms
Consider a (continuous-time) symmetric Markovian jump process on a metric measure space. If the underlying metric measure space satisfies the volume-doubling and reverse-volume-doubling properties, then it is known that two-sided heat kernel estimates and the parabolic Harnack inequality are both stable under bounded perturbations of the jumping measure. However, the reverse-volume-doubling condition fails if the metric measure space is a graph (or more generally, if it contains any atoms). We generalize these previously known stability results to spaces that satisfy what may be thought of as "reverse-volume-doubling at sufficiently large scales". In particular, we show that heat kernel estimates and the parabolic Harnack inequality are both stable for symmetric jump processes on graphs (with the usual graph metric) that have infinite diameter and satisfy the volume-doubling property.