On the fundamental solution for degenerate Kolmogorov equations with rough coefficients

Abstract

The aim of this work is to prove the existence of a fundamental solution associated to the Kolmogorov equation L u = f with measurable coefficients in the dilation invariant case. Moreover, we prove Gaussian upper and lower bounds for it, and other related properties.

Optimal regularity for degenerate Kolmogorov equations with rough coefficients

“…For Kolmogorov operators with coefficients that are Hölder continuous in both space and time, the study of the existence of a fundamental solution goes back to the early papers [31], [9], [30] and [19]. A modern and more natural approach based on the Lie group theory was developed by [28], [5], [1] and [24]. Applications to the martingale problem for some degenerate diffusion processes are given in [20] and [21].…”