Trend to equilibrium and hypoelliptic regularity for the relativistic Fokker-Planck equation

We consider the relativistic, spatially inhomogeneous Fokker-Planck equation with an external confining potential. We prove the exponential time decay of solutions towards the global equilibrium in weighted $L^2$ and Sobolov spaces. Our result holds for a wide class of external potentials and the estimates on the rate of convergence are explicit and constructive. Moreover, we prove that the associated semigroup of the equation has hypoelliptic regularizing properties and we obtain explicit rates on this regularization. The technique is based on the construction of suitable Lyapunov functionals.