A Global Lower Bound for the Fundamental Solution of Kolmogorov-Fokker-Planck Equations

Abstract

The main result of this paper is a global lower bound for the fundamental solution Γ of the ultraparabolic differential operator

where the a _{i , j} 's and their first derivatives are Hölder continuous functions and 0 < p ₀ < N. The bound will follow from a local estimate of Γ and a Harnack inequality for non-negative solutions of Lu = 0, by exploiting the invariance of the Harnack inequality with respect to suitable translation and dilation groups. For non-degenerate parabolic operators, our methods and results generalize those of Aronson & Serrin [1].