Harnack inequality, if it holds, is a useful tool in analysis and probability theory.

In this talk, I will discuss scale invariant elliptic Harnack inequality for general diffusions, or equivalently, for general differential operators on metric measure spaces, and show that it is stable under form-comparable perturbations for strongly local Dirichlet forms on complete locally compact separable  metric spaces that satisfy metric doubling property. 

Based on Joint work with Martin Barlow and Mathav Murugan.