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Where does randomness lead in spacetime?

Published online by Cambridge University Press:  22 May 2008

Ismael Bailleul  and
Albert Raugi
Ismael Bailleul
Affiliation:
Cambridge University; i.bailleul@statslab.cam.ac.uk
Albert Raugi
Affiliation:
Université Rennes 1; raugi@univ-rennes1.fr
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Abstract

We provide an alternative algebraic and geometric approach to the results of [I. Bailleul, Probab. Theory Related Fields141 (2008) 283–329] describing the asymptotic behaviour of the relativistic diffusion.

Keywords

Random walks on groupsPoisson boundaryspecial relativitycausal boundary.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 16 - 52
Copyright
© EDP Sciences, SMAI, 2010

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References

Ancona, A., Théorie du potentiel sur les graphes et les variétés. École d'été de Probabilités de Saint-Flour XVIII, 1988. Lect. Notes Math. 1427 (1990) 1–112. Springer, Berlin.
Applebaum, D., Compound Poisson processes and Lévy processes in groups and symmetric spaces. J. Theoret. Probab. 13 (2000) 383–425. CrossRef
Applebaum, D. and Kunita, H., Lévy flows on manifolds and Lévy processes on Lie groups. J. Math. Kyoto Univ. 33 (1993) 1103–1123. CrossRef
Bailleul, I., Poisson boundary of a relativistic diffusion. Probab. Theory Related Fields 141 (2008) 283–329. CrossRef
Baldi, P. and Chaleyat-Maurel, M., Sur l'équivalent du module de continuité des processus de diffusion, in Séminaire de Probabilités, XXI. Lect. Notes Math. 1247 (1987) 404–427. Springer, Berlin. CrossRef
J.K. Beem, P.E. Ehrlich and K.L. Easley, Global Lorentzian geometry, volume 202 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York, second edition (1996).
A.N. Borodin and P. Salminen, Handbook of Brownian motion – facts and formulae. Probability and its Applications. Birkhäuser Verlag, Basel. Second edition (2002).
Derriennic, Y., Lois “zéro ou deux” pour les processus de Markov. Applications aux marches aléatoires. Ann. Inst. H. Poincaré Sect. B (N.S.) 12 (1976) 111–129.
Dudley, R.M., Lorentz-invariant Markov processes in relativistic phase space. Ark. Mat. 6 (1966) 241–268. CrossRef
Dudley, R.M., Asymptotics of some relativistic Markov processes. Proc. Natl. Acad. Sci. USA 70 (1973) 3551–3555. CrossRef
C. Frances, Géométrie et dynamique Lorentzienne conformes. École Normale Supérieure de Lyon (2002).
Geroch, R., Kronheimer, E.H., and Roger Penrose, Ideal points in space-time. Proc. Roy. Soc. Lond. Ser. A 327 (1972) 545–567. CrossRef
Grigor'yan, A., Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N.S.) 36 (1999) 135–249. CrossRef
Y. Guivarc'h. Une loi des grands nombres pour les groupes de Lie. In Séminaire de Probabilités, I . Exposé No. 8. Dépt. Math. Informat., Univ. Rennes, France (1976).
Hurd, T.R., The projective geometry of simple cosmological models. Proc. Roy. Soc. Lond. Ser. A 397 (1985) 233–243. CrossRef
N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes. North-Holland Mathematical Library, Vol. 24. North-Holland Publishing Co., Amsterdam, second edition (1989).
Karpelevič, F.I., Tutubalin, V.N. and Šur, M.G., Limit theorems for compositions of distributions in the Lobačevskiĭ plane and space. Teor. Veroyatnost. i Primenen. 4 (1959) 432–436.
M. Liao, Lévy processes in Lie groups, volume 162 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2004).
J. Neveu, Mathematical foundations of the calculus of probability. Translated by Amiel Feinstein. Holden-Day Inc., San Francisco, Californie (1965).
B. O'Neill, Semi-Riemannian geometry. With applications to relativity, volume 103 of Pure Appl. Math. Academic Press Inc. (Harcourt Brace Jovanovich Publishers), New York (1983).
R.G. Pinsky, Positive harmonic functions and diffusion, volume 45 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995).
J.-J. Prat, Étude asymptotique et convergence angulaire du mouvement brownien sur une variété à courbure négative. C. R. Acad. Sci. Paris Sér. A-B 280 Aiii (1975) A1539–A1542.
A. Raugi, Fonctions harmoniques sur les groupes localement compacts à base dénombrable. Bull. Soc. Math. France, Mémoire 54 (1977) 5–118.
A. Raugi, Périodes des fonctions harmoniques bornées. In Seminar on Probability, Rennes, 1978 (French). Exposé No. 10. Univ. Rennes, France (1978).