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In this paper, we determine the Poisson boundary of the relativistic Brownianmotion in two classes of Lorentzian manifolds, namely model manifolds ofconstant scalar curvature and Robertson--Walker space-times, the latterconstituting a large family of curved manifolds. Our objective is two fold: onthe one hand, to understand the interplay between the geometry at infinity ofthese manifolds and the asymptotics of random sample paths, in particular tocompare the stochastic compactification given by the Poisson boundary toclassical purely geometric compactifications such as the conformal or causalboundaries. On the other hand, we want to illustrate the power of thed\'evissage method introduced by the authors in [AT16], method which we show tobe particularly well suited in the geometric contexts under consideration here.

On the Poisson boundary of the relativistic Brownian motion