Heat kernel estimates for nonlocal kinetic operators

TL;DR

This paper derives sharp heat kernel estimates and gradient bounds for nonlocal kinetic operators using probabilistic methods, addressing a key problem in nonlocal kinetic theory.

Contribution

It provides explicit two-sided heat kernel estimates and gradient bounds for nonlocal kinetic operators, including more general stable-like operators, advancing the understanding of their fundamental properties.

Findings

Sharp two-sided heat kernel estimates derived

Logarithmic gradient estimates established

Results apply to general non-symmetric stable-like 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 $$ \Delta^{\alpha/2}_v + v \cdot \nabla_x, \quad \alpha \in (0, 2),\ (x,v)\in {\mathbb R}^{d}\times{\mathbb R}^d,$$ 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.