Abstract: The notion of conformal walk dimension serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that the finiteness of the conformal walk dimension characterizes the elliptic Harnack inequality.

Roughly speaking, the conformal walk dimension is the infimum of all possible values of the walk dimension that can be attained by a time-change of the process and by a quasisymmetric change of the metric. Two natural questions arise (a) What are the possible values of the conformal walk dimension? (b) When is the infimum attained? In this talk, I will explain the answer to (a), and mention partial progress towards (b).

This talk is based on joint work with Naotaka Kajino.