Fundamental solutions for Kolmogorov-Fokker-Planck operators with time-depending measurable coefficients

Abstract

We consider a Kolmogorov-Fokker-Planck operator of the kind studied by Lanconelli-Polidoro in [Rend. Sem. Mat. Univ. Politec. Torino 52 (1994)], where the leading coefficients $a_{ij}$, instead of being constant, are bounded measurable functions of t. We construct an explicit fundamental solution for this operator, study its property, show a comparison result between this function and the fundamental solution of some model operators with constant $a_{ij}$, and show the unique solvability of the Cauchy problem under various assumptions on the initial datum.