Let be a metric measure space with a local regular Dirichlet form. We give necessary and sufficient conditions for a parabolic Harnack inequality with global space-time scaling exponent to hold. We show that this parabolic Harnack inequality is stable under rough isometries. As a consequence, once such a Harnack inequality is established on a metric measure space, then it holds for any uniformly elliptic operator in divergence form on a manifold naturally defined from the graph approximation of the space.
"Stability of parabolic Harnack inequalities on metric measure spaces." J. Math. Soc. Japan 58 (2) 485 - 519, April, 2006. https://doi.org/10.2969/jmsj/1149166785