Stability of parabolic Harnack inequalities on metric measure spaces

Let (X, d, mu) 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 beta >= 2 to hold. We show that this parabolic Harnack inequality is stable under rough isometrics. 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.