Abstract: 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. This talk is based on a joint work with Xicheng Zhang(Beijing Institute of Technology).