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.
DOI : 10.1051/ps/2015003
Mots-clés : Relativistic diffusion, lorentzian manifolds, poisson boundary, causal boundary
@article{PS_2015__19__502_0,
author = {Angst, J\"urgen},
title = {Poisson boundary of a relativistic diffusion in curved space-times: an example},
journal = {ESAIM: Probability and Statistics},
pages = {502--514},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2015003},
zbl = {1333.60168},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2015003/}
}
TY - JOUR AU - Angst, Jürgen TI - Poisson boundary of a relativistic diffusion in curved space-times: an example JO - ESAIM: Probability and Statistics PY - 2015 SP - 502 EP - 514 VL - 19 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2015003/ DO - 10.1051/ps/2015003 LA - en ID - PS_2015__19__502_0 ER -
%0 Journal Article %A Angst, Jürgen %T Poisson boundary of a relativistic diffusion in curved space-times: an example %J ESAIM: Probability and Statistics %D 2015 %P 502-514 %V 19 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2015003/ %R 10.1051/ps/2015003 %G en %F PS_2015__19__502_0
Angst, Jürgen. Poisson boundary of a relativistic diffusion in curved space-times: an example. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 502-514. doi : 10.1051/ps/2015003. http://archive.numdam.org/articles/10.1051/ps/2015003/
and , The causal boundary of product spacetimes. Gen. Relativ. Gravitation 39 (2007) 1697–1718. | Zbl
and , Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker space-times. Math. Proc. Cambridge Philos. Soc. 143 (2007) 703–729. | Zbl
J. Angst, Asymptotic behavior of a relativistic diffusion in Robertson-Walker space-times. To appear in Ann. Inst. Henri Poincaré. Preprint arXiv:1405.0142 (2014).
J. Angst and C. Tardif, Dévissage of a Poisson boundary under equivariance and regularity conditions. Séminaire de probabilités de Strasbourg. Preprint arXiv:1311.4428 (2015).
J. Angst and C. Tardif, Poisson boundary of a relativistic Markov processes. Work in progress (2015).
, and , Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature. Mathematische Zeitschrifft 263 (2009) 369–409. | Zbl
, Poisson boundary of a relativistic diffusion. Probab. Theory Related Fields 141 (2008) 283–329. | Zbl
, Lorentz-invariant Markov processes in relativistic phase space. Ark. Mat. 6 (1966) 241–268. | MR | Zbl
, Asymptotics of some relativistic Markov processes. Proc. Natl. Acad. Sci. USA 70 (1973) 3551–3555. | MR | Zbl
and , Relativistic diffusions and Schwarzschild geometry. Comm. Pure Appl. Math. 60 (2007) 187–251. | MR | Zbl
, and , Ideal points in space-time. Roy. Soc. London Proc. Ser. A 327 (1972) 545–567. | MR | Zbl
S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time. Vol. I of Camb. Monogr. Math. Phys. Cambridge University Press, London (1973). | MR | Zbl
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusions Processes. North Holland, 2nd edition (1989). | MR | Zbl
, Lyapunov spectrum of a relativistic stochastic flow in the Poincaré group. Stoch. Dyn. 14 (2014) 1450013. | MR | Zbl
S. Weinberg, Gravitation and Cosmology. John Wiley and Sons (1972).
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