The De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusion

I will present some recent results for weak solutions to a wide class of kinetic integral equations, where the diffusion term in velocity is an integro-differential operator having nonnegative kernel of fractional order $s$ in $(0,1)$ with merely measurable coefficients. In particular, I will focus on Harnack-type inequalities for nonnegative weak solutions that does not require the usual a priori boundedness.
The talk is based on a series of papers by Anceschi, Kassmann, Piccinini, Weidner, and myself.