Stability of parabolic Harnack inequalities

Abstract

Let (G, E) be a graph with weights {a xy } for which a parabolic Harnack inequality holds with space-time scaling exponent β ≥ 2. Suppose {a xy } is another set of weights that are comparable to {a xy }. We prove that this parabolic Harnack inequality also holds for (G, E) with the weights {a xy }. We also give stable necessary and sufficient conditions for this parabolic Harnack inequality to hold.