Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Jürgen Angst

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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.

Résumé

Nous déterminons le comportement asymptotique en temps long d’une diffusion relativiste à valeurs dans le fibré tangent unitaire d’un espace de Robertson–Walker. On montre en particulier qu’au voisinage du temps d’explosion de la diffusion, sa projection sur la variété de base converge presque sûrement vers un point aléatoire de la frontière causale et nous décrivons le comportement du vecteur tangent au voisinage de ce point limite.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 376-411.

Dates
Received: 19 June 2013
Revised: 1 April 2014
Accepted: 29 April 2014
First available in Project Euclid: 6 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089273

Digital Object Identifier
doi:10.1214/14-AIHP626

Mathematical Reviews number (MathSciNet)
MR3449307

Zentralblatt MATH identifier
1344.58017

Subjects
Primary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]
Secondary: 53C50: Lorentz manifolds, manifolds with indefinite metrics 83F05: Cosmology

Keywords
Brownian Motion Relativistic diffusion Robertson–Walker space–times Causal boundary

Citation

Angst, Jürgen. Asymptotic behavior of a relativistic diffusion in Robertson–Walker space–times. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 376--411. doi:10.1214/14-AIHP626. https://projecteuclid.org/euclid.aihp/1452089273


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