Nash's $G$ bound for the Kolmogorov equation

TL;DR

This paper establishes Nash's $G$ bound for the Kolmogorov equation with rough coefficients, leading to a new proof of the Harnack inequality in the kinetic setting.

Contribution

It extends Nash's $G$ bound to the Kolmogorov equation with rough coefficients using kinetic trajectories, providing an alternative proof of the Harnack inequality.

Findings

Nash's $G$ bound is proven for the Kolmogorov equation.

The fundamental solution's sharp lower bound is recovered.

An alternative proof of the Harnack inequality is provided.

Abstract

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.