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

We consider a Kolmogorov-Fokker-Planck operator of the kind: (equation presented) where n ai j (t) oq i; j=1 is a symmetric uniformly positive matrix on Rq, q ≤ N, of bounded measurable coefficients defined for t 2 R and the matrix B = n bi j oN i; j=1 satisfies a structural assumption which makes the corresponding operator with constant ai j hypoelliptic. 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 ai j, and show the unique solvability of the Cauchy problem for L under various assumptions on the initial datum.