Heat kernel estimates for kinetic SDEs with drifts being unbounded and in Kato's class

Abstract

In this paper we investigate the existence and uniqueness of weak solutions for kinetic stochastic differential equations with H\"older diffusion and unbounded singular drifts in Kato's class. Moreover, we also establish sharp two-sided estimates for the density of the solution. In particular, the drift $b$ can be in the mixed $L^q_tL^{p_1}_{x_1}L^{p_2}_{x_2}$ space with $\frac2q+\frac{d}{p_1}+\frac{3d}{p_2}<1$. As an application, we show the existence and uniqueness of weak solution to a second order singular interacting particle system in ${\mathbb R}^{d N}$.