Zhen-Qing Chen, UW

Tuesday, February 7, 2023 - 1:30pm to 3:30pm

PDL C-401

Elliptic Harnack inequality, if it holds, is a very useful tool in analysis and in probability theory. An important question

is whether elliptic Harnack inequality is stable under bounded perturbation. In this talk, I will explain how probabilistic

ideas can be used to address this open problem.

In the first part of the talk, I will present a gentle introduction of elliptic as well as parabolic Harnack inequalities and some of their history.

In the second part of the talk, I will discuss scale invariant elliptic Harnack inequality for symmetric differential operators on metric measure space such as manifolds, graphs and fractals, or equivalently, for symmetric differential operators on metric measure space. I will then show that the elliptic Harnack inequality is stable under form-comparable perturbation for strongly local Dirichlet forms on complete locally compact separable metric spaces that satisfy metric doubling property.

Based on a joint work with Martin Barlow and Mathav Murugan.