In this talk, we will discuss parabolic and elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We will present stable equivalent characterizations of parabolic Harnack inequalitiesin terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincare inequalities. In particular, we establish the connection between parabolic Harnack inequalities, elliptic Harnack inequalities, and two-sided heat kernel estimates, as well as with the Holder regularity of parabolic functions for symmetric non-local Dirichlet forms. Stability of elliptic Harnack inequalities will also be discussed, if time permits. Based on joint work with Takashi Kumagai and Jian Wang.

Given a locally compact separable ultra-metric space with a Radon measure, we construct a class of symmetric Markov semigroups and the corresponding Markov processes. We obtain upper and lower bounds of its transition density and Green function, give a transience criterion, estimate its moments and describe the Markov generator. In particular, our results apply on the field of p-adic numbers, where we obtain new results about the Taibleson and Vladimirov Laplacians.

We prove necessary and sufficient conditions for stable-like estimates of the heat kernel for jump type Dirichlet forms on metric measure spaces. The conditions are given in terms of the volume growth function, jump kernel and a generalized capacity. Joint with Alexander Grigor'yan and Eryan Hu (Bielefeld).