Trends to equilibrium for a class of relativistic diffusions
Résumé
A large class C of relativistic diffusions with values in the phase-space of special relativity was introduced by Angst and Franchi [J. Math. Phys. 48(8), 083101 (2007)] in order to answer some open questions concerning the asymptotic behavior of two examples of such processes. In particular, the equilibrium measures of these diffusions were explicitly computed, and their hydrodynamic limit was shown to be Brownian. In this paper, we address the question of the trends to equilibrium of the momentum component of the diffusions of the whole class C. We show the existence of a spectral gap using the Lyapounov method and deduce the exponential decay of the distance to equilibrium in L(2)-norm and in total variation. A similar result was obtained recently by Calogero [e-print arXiv:1009.5086v2] for a particular process of the class C.