  In this paper, we determine the Poisson boundary of the relativistic Brownian
motion in two classes of Lorentzian manifolds, namely model manifolds of
constant scalar curvature and Robertson--Walker space-times, the latter
constituting a large family of curved manifolds. Our objective is two fold: on
the one hand, to understand the interplay between the geometry at infinity of
these manifolds and the asymptotics of random sample paths, in particular to
compare the stochastic compactification given by the Poisson boundary to
classical purely geometric compactifications such as the conformal or causal
boundaries. On the other hand, we want to illustrate the power of the
d\'evissage method introduced by the authors in [AT16], method which we show to
be particularly well suited in the geometric contexts under consideration here.
