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

TL;DR

This paper establishes a Lorentz-invariant Harnack inequality and asymptotic lower bounds for solutions of a class of relativistic kinetic operators, advancing understanding of relativistic stochastic processes.

Contribution

It introduces a Lorentz-invariant Harnack inequality and derives precise asymptotic lower bounds for solutions of relativistic Fokker-Planck operators, a novel contribution in the field.

Findings

Proved a Lorentz-invariant Harnack inequality.

Derived asymptotic lower bounds for solutions.

Established lower bounds for the associated stochastic process density.

Abstract

We consider a class of second order degenerate kinetic operators $\mathscr{L}$ in the framework of special relativity. We first describe $\mathscr{L}$ as an H\"ormander 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 $\mathscr{L} f = 0$. As a consequence we obtain a lower bound for the density of the relativistic stochastic process associated to $\mathscr{L}$.