Harnack inequality for a class of Kolmogorov-Fokker-Planck equations in non-divergence form

TL;DR

This paper establishes invariant Harnack inequalities for specific Kolmogorov-type equations in non-divergence form, leveraging Lie group invariance and conditions on coefficients to extend regularity results.

Contribution

It introduces new invariant Harnack inequalities for Kolmogorov equations with coefficients satisfying Cordes-Landis or continuity conditions, under Lie group invariance.

Findings

Proves Harnack inequalities for non-divergence Kolmogorov equations.

Extends regularity theory to equations with less restrictive coefficient conditions.

Utilizes Lie group invariance to establish inequalities.

Abstract

We prove invariant Harnack inequalities for certain classes of non-divergence form equations of Kolmogorov type. The operators we consider exhibit invariance properties with respect to a homogeneous Lie group structure. The coefficient matrix is assumed either to satisfy a Cordes-Landis condition on the eigenvalues, or to admit a uniform modulus of continuity.