We use probabilistic techniques to derive sharp and explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator \( \Delta ^{\alpha /2}_v + v \cdot \nabla _x, \quad \alpha \in (0, 2),\ (x,v)\in {\mathbb {R}}^{d}\times {\mathbb {R}}^d, \) where \( \Delta ^{\alpha /2}_v \) 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.
