Following the work of Moser, as well as de Giorgi and Nash, Harnack inequalities have proved to be a powerful tool in PDE as well as in probability. In the early 1990s Grigor’yan and Saloff-Coste gave a characterisation of the parabolic Harnack inequality (PHI). This characterisation implies that the PHI is stable under bounded perturbation of weights, as well as rough isometries. In this talk we prove the stability of the EHI. The proof uses the concept of a quasi symmetric transformation of a metric space, and the introduction of these ideas to Markov processes suggests a number of new problems.

This is joint work with Mathav Murugan (UBC).

(This talk is part of the 2018 Finger Lakes Probability Seminar.)