  We investigate the Cauchy problem and the diffusion asymptotics for a
spatially inhomogeneous kinetic model associated to a nonlinear Fokker-Planck
operator. We derive the global well-posedness result with instantaneous
smoothness effect, when the initial data lies below a Maxwellian. The proof
relies on the hypoelliptic analog of classical parabolic theory, as well as a
positivity-spreading result based on the Harnack inequality and barrier
function methods. Moreover, the scaled equation leads to the fast diffusion
flow under the low field limit. The relative phi-entropy method enables us to
see the connection between the overdamped dynamics of the nonlinearly coupled
kinetic model and the correlated fast diffusion. The global in time
quantitative diffusion asymptotics is then derived by combining entropic
hypocoercivity, relative phi-entropy and barrier function methods.
