Nash's $G$ bound for the Kolmogorov equation

TL;DR

This work extends Nash’s $G$ bound to the Kolmogorov equation with rough diffusion by introducing critical kinetic trajectories, enabling a trajectory-based $L^1$-log estimate for logarithms of supersolutions. The main result is a universal lower bound for the fundamental solution $\Gamma$, obtained through a kinetic version of Nash’s argument and a careful combination of scaling, Galilean invariance, and semigroup properties, which also yields a kinetic Harnack inequality. The approach provides an alternative to the Moser–Fabes–Stroock route by deriving a sharp Gaussian-type lower bound directly from the $G$ bound, without requiring a spatial Poincaré inequality with Gaussian weight. Together with the known upper bounds and the recently established fundamental solution existence for rough coefficients, the results reinforce the kinetic De Giorgi–Nash–Moser theory and offer robust tools for quantitative bounds on kinetic Fokker–Planck-type equations.

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.

Theorems & Definitions (13)

• Lemma 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 3.1
• proof
• proof : Proof of Lemma \ref{['lem:nashG']}
• Remark 3.2
• Remark 3.3
• Corollary 3.4
• proof