Fundamental solution of kinetic Fokker-Planck operator with anisotropic nonlocal dissipativity

By using the probability approach (the Malliavin calculus), we prove the existence of smooth fundamental solutions for degenerate kinetic Fokker-Planck equation with anisotropic nonlocal dissipativity, where the dissipative term is the generator of an anisotropic L'evy process and the drift term is allowed to be cubic growth.