Heat kernel estimates for nonlocal kinetic operators

In this paper, we employ probabilistic techniques to derive sharp, 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 \( \Delta^{\alpha/2}_v \) represents the fractional Laplacian acting on the velocity variable \(v\). Additionally, we establish logarithmic gradient estimates with respect to both the spatial variable \(x\) and the velocity variable \(v\). In fact, the estimates are developed for more general non-symmetric stable-like operators, demonstrating explicit dependence on the lower and upper bounds of the kernel functions. These results, in particular, provide a solution to a fundamental problem in the study of \emph{nonlocal} kinetic operators.