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

We consider a degenerate Kolmogorov-Fokker-Planck operator in non-divergence form bounded measurable coefficients. We assume that the drift term is a linear function of the space variable that makes hypoelliptic the corresponding operator with constant coefficients. We construct an explicit fundamental solution Γ for L, study its properties, show a comparison result between Γ and the fundamental solution of some model operators with constant coefficients, and show the unique solvability of the Cauchy problem for L under various assumptions on the initial datum.