A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients

TL;DR

This paper presents a geometric formulation of the Harnack inequality for degenerate Kolmogorov equations with rough coefficients, leading to a strong maximum principle for weak solutions.

Contribution

It provides a geometric reinterpretation of the Harnack inequality for a class of degenerate PDEs with measurable coefficients, extending previous results.

Findings

Established a geometric Harnack inequality for degenerate Kolmogorov equations.

Derived a strong maximum principle as a corollary.

Applicable to PDEs with rough, measurable coefficients.

Abstract

We consider weak solutions of degenerate second order partial differential equations of Kolmogorov-Fokker-Planck type with measurable coefficients in divergence form. We give a geometric statement of the Harnack inequality recently proven by Golse, Imbert, Mouhot and Vasseur. As a corollary we obtain a strong maximum principle.