Harnack inequality and asymptotic lower bounds for the relativistic Fokker-Planck operator

We consider a class of second order degenerate kinetic operators $\mathcal{L}$ in the framework of special relativity. We first describe $\mathcal{L}$ as an Hörmander operator which is invariant with respect to Lorentz transformations. Then we prove a Lorentz-invariant Harnack type inequality, and we derive accurate asymptotic lower bounds for positive solutions to $\mathcal{L}f=0$. As a consequence we obtain a lower bound for the density of the relativistic stochastic process associated to $\mathcal{L}$.