Degenerate Fokker–Planck equations: Bismut formula, gradient estimate and Harnack inequality

By constructing successful couplings for degenerate diffusion processes, explicit derivative formula and Harnack type inequalities are presented for solutions to a class of degenerate Fokker–Planck equations on Rm×Rd. The main results are also applied to the study of gradient estimate, entropy/transportation-cost inequality and heat kernel inequalities.