  We study a simple microscopic model for the one-dimensional stochastic motion
of a (non)relativistic Brownian particle, embedded into a heat bath consisting
of (non)relativistic particles. The stationary momentum distributions are
identified self-consistently (for both Brownian and heat bath particles) by
means of two coupled integral criteria. The latter follow directly from the
kinematic conservation laws for the microscopic collision processes, provided
one additionally assumes probabilistic independence of the initial momenta. It
is shown that, in the nonrelativistic case, the integral criteria do correctly
identify the Maxwellian momentum distributions as stationary (invariant)
solutions. Subsequently, we apply the same criteria to the relativistic case.
Surprisingly, we find here that the stationary momentum distributions differ
slightly from the standard J\"uttner distribution by an additional prefactor
proportional to the inverse relativistic kinetic energy.
