Poisson boundary of a relativistic diffusion in curved space-times: an example

TL;DR

This paper investigates the long-term behavior of a relativistic diffusion process in curved space-times, showing that its Poisson boundary aligns with the causal boundary of the manifold, especially in a specific cosmological model.

Contribution

It establishes a connection between the Poisson boundary of relativistic diffusions and the causal boundary in curved Lorentzian manifolds, providing a detailed analysis in a Robertson-Walker space-time.

Findings

Poisson boundary identified with the causal boundary

Detailed analysis in a spatially flat, expanding universe

Results applicable to relativistic diffusions in curved space-times

Abstract

We study in details the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a curved Lorentzian manifold, namely a spatially flat and fast expanding Robertson-Walker space-time. We prove in particular that the Poisson boundary of the diffusion can be identified with the causal boundary of the underlying manifold.