Long time behavior of non-autonomous Fokker-Planck equations and the cooling of granular gases

We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases as introduced J. J. Brey, J. Dufty and A. Santos (1999), by showing that the Homogenous Cooling State attracts any solution at an algebraic rate.