On the Poisson boundary of the relativistic Brownian motion

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 set of exit points of the process to classical purely geometric compactifications such as the conformal or causal boundaries. On the other hand, we want to illustrate the power of the devissage method introduced by the authors (in Seminaire de Probabilites XLVIII (2016) 199-229 Springer), method which we show to be particularly well suited in the geometric contexts under consideration here.