Heat kernel estimates for nonlocal kinetic operators

We use probabilistic techniques to derive sharp and explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator (Formula presented.) where Δvα/2 denotes the fractional Laplacian acting on the velocity variable v. We also establish logarithmic gradient estimates with respect to both the spatial variable x and the velocity variable v. In fact, our estimates are obtained for more general nonsymmetric stable-like operators and make the dependence on the lower and upper bounds of the kernel explicit. These results provide, in particular, a solution to a fundamental problem in the study of nonlocal kinetic operators.