Harnack estimates for nonlocal drift-diffusion equations

A set of pointwise estimates are established for local solutions to nonlocal diffusion equations with a drift term. In particular, our Harnack estimates are the first ones for such equations, and our Hölder regularity refines certain known result in several aspects. The approach is measure theoretical in the spirit of DeGiorgi classes. It yields novel nonlocal weak Harnack estimates in the elliptic case as well.