Upper bounds for fundamental solutions to non-local diffusion equations with divergence free drift

We consider some non-local diffusion equations in the presence of a divergence free drift term, where the diffusion operators are related to certain Dirichlet forms of jump type. We derive pointwise upper bounds for fundamental solutions of the equations under weak assumptions for the velocity of the drift term. Our class of the velocity includes functions with the scale-critical regularity and some growing functions at spatial infinity. (C) 2013 Elsevier Inc. All rights reserved.