# On the Poisson boundary of the relativistic Brownian motion

## TL;DR

This paper characterizes the Poisson boundary of relativistic Brownian motion on specific Lorentzian manifolds, linking geometric infinity with stochastic behavior, and demonstrates the effectiveness of a novel decomposition method in these settings.

## Contribution

It identifies the Poisson boundary for relativistic Brownian motion on model and Robertson--Walker manifolds, illustrating the method's power in curved spacetime geometries.

## Key findings

- Poisson boundary matches geometric boundaries in studied manifolds
- The devissage method effectively analyzes stochastic asymptotics in Lorentzian geometry
- Comparison between stochastic and classical geometric compactifications

## Abstract

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.