| Abstract: |
| We are concerned with the study of a second order degenerate kinetic operator in the framework of special relativity. In classical mechanics, the Fokker-Planck equation describes the diffusion of particles in the phase space, and satisfies a weak H\ormander condition. Despite of its many physical applilcations, an undesirable feature of its classical diffusion term is that it operates with infinite velocity, which is in conflict with special relativity. We consider the differential operator
\begin{equation*}
\mathscr{L} f (p,y,t) = {\sqrt{p^{2} + 1}}\, \tfrac{\partial}{\partial p}
\left({\sqrt{p^{2} + 1}}\, \tfrac{\partial f}{\partial p} \right)
- p \tfrac{\partial f}{\partial y} - \sqrt{p^{2} + 1} \,
\tfrac{\partial f}{\partial t}
\end{equation*}
which has the additional property of being invariant with respect to the Lorentz change of variable. Furthermore, $\mathscr{L}$ can be written in the H\ormander`s form $\mathscr{L} = X^2 + Y$. This paves the way to a systematic study of the operator $\mathscr{L}$ in the framework of degenerate H\ormander`s operators in Lie groups. Our main results are a Lorentz-invariant Harnack type inequality, and an accurate asymptotic lower bound for the fundamental solution to $\mathscr{L} f = 0$. |
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