Nash's $G$ bound for the Kolmogorov equation

We prove Nash’s $G$ bound for the Kolmogorov equation with rough coefficients. Our proof is inspired by the treatment of the parabolic problem by Nash (1958) and Fabes and Stroock (1986). To transfer their ideas to the kinetic setting, we employ critical kinetic trajectories. From Nash’s $G$ bound, we recover the sharp lower bound on the fundamental solution and thus provide an alternative proof of the Harnack inequality for the Kolmogorov equation.