Optimal regularity for degenerate Kolmogorov equations with rough coefficients

TL;DR

This paper proves the existence and optimal regularity of fundamental solutions for a class of degenerate Kolmogorov equations with rough coefficients, advancing understanding of their regularity and Gaussian behavior.

Contribution

It introduces a generalized notion of strong solutions and establishes fundamental solution existence, regularity, and Gaussian estimates for degenerate equations with measurable and H"older continuous coefficients.

Findings

Existence of fundamental solutions for the class of equations.

Optimal H"older regularity of solutions.

Gaussian estimates for the fundamental solutions.

Abstract

We consider a class of degenerate equations satisfying a parabolic H\"ormander condition, with coefficients that are measurable in time and H\"older continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal H\"older regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin-type diffusions.