Asymptotic behavior of a relativistic diffusion in Robertson-Walker space-times

TL;DR

This paper analyzes the long-term behavior of a relativistic diffusion process in Robertson-Walker space-times, showing convergence to a boundary point and describing tangent vector behavior near this limit.

Contribution

It provides a detailed description of the asymptotic behavior of relativistic diffusions in cosmological models, including convergence to the causal boundary.

Findings

Diffusion's projection converges to a random boundary point

Behavior of tangent vectors near the boundary is characterized

Results apply to the long-time dynamics in cosmological space-times

Abstract

We determine the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a Robertson-Walker space-time. We prove in particular that when approaching the explosion time of the diffusion, its projection on the base manifold almost surely converges to a random point of the causal boundary and we also describe the behavior of the tangent vector in the neighborhood of this limiting point.