Heat kernel estimates for nonlocal kinetic operators

Abstract

In this paper, we employ probabilistic techniques to derive sharp, explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator $$\hat{I}{”}_v^{\hat{I}\pm /2}+v\cdot {\nabla }_x,\hat{I}\pm \in (0,2),(x,v)\in {\mathbb{R}}^d\times {\mathbb{R}}^d,$$ where $\hat{I}{”}_v^{\hat{I}\pm /2}$ 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.

25pages, update the references and correct several typos