Trends to equilibrium for a class of relativistic diffusions

We address the question of the trends to equilibrium for a large class C of relativistic diffusions. We show the existence of a spectral gap using the Lyapounov method and deduce the exponential decay of the distance to equilibrium in L2-norm and in total variation. A similar result was obtained recently in arXiv:1009.5086 for a particular process of the class C.