  We consider an infinite system of Brownian motions which interact through a
given Brownian motion being reflected from its left neighbor. Earlier we
studied this system for deterministic periodic initial configurations. In this
contribution we consider initial configurations distributed according to a
Poisson point process with constant intensity, which makes the process
space-time stationary. We prove convergence to the Airy process for stationary
the case. As a byproduct we obtain a novel representation of the
finite-dimensional distributions of this process. Our method differs from the
one used for the TASEP and the KPZ equation by removing the initial step only
after the limit $t\to\infty$. This leads to a new universal cross-over process.
