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 ${\mathbb{R}}^m\times {\mathbb{R}}^d$. The main results are also applied to the study of gradient estimate, entropy/transportation-cost inequality and heat kernel inequalities.