Where does randomness lead in spacetime ?
ESAIM: Probability and Statistics, Volume 14 (2010), pp. 16-52.

We provide an alternative algebraic and geometric approach to the results of [I. Bailleul, Probab. Theory Related Fields 141 (2008) 283-329] describing the asymptotic behaviour of the relativistic diffusion.

DOI: 10.1051/ps:2008021
Classification: 60B99, 60J50, 60J45, 83A05
Keywords: random walks on groups, Poisson boundary, special relativity, causal boundary
     author = {Bailleul, Ismael and Raugi, Albert},
     title = {Where does randomness lead in spacetime ?},
     journal = {ESAIM: Probability and Statistics},
     pages = {16--52},
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     volume = {14},
     year = {2010},
     doi = {10.1051/ps:2008021},
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     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2008021/}
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Bailleul, Ismael; Raugi, Albert. Where does randomness lead in spacetime ?. ESAIM: Probability and Statistics, Volume 14 (2010), pp. 16-52. doi : 10.1051/ps:2008021. http://www.numdam.org/articles/10.1051/ps:2008021/

[1] A. Ancona, 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. | Zbl

[2] D. Applebaum, Compound Poisson processes and Lévy processes in groups and symmetric spaces. J. Theoret. Probab. 13 (2000) 383-425. | Zbl

[3] D. Applebaum and H. Kunita, Lévy flows on manifolds and Lévy processes on Lie groups. J. Math. Kyoto Univ. 33 (1993) 1103-1123. | Zbl

[4] I. Bailleul, Poisson boundary of a relativistic diffusion. Probab. Theory Related Fields 141 (2008) 283-329. | Zbl

[5] P. Baldi and M. Chaleyat-Maurel, 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. | Numdam | Zbl

[6] 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). | Zbl

[7] A.N. Borodin and P. Salminen, Handbook of Brownian motion - facts and formulae. Probability and its Applications. Birkhäuser Verlag, Basel. Second edition (2002). | Zbl

[8] Y. Derriennic, 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. | Numdam | Zbl

[9] R.M. Dudley, Lorentz-invariant Markov processes in relativistic phase space. Ark. Mat. 6 (1966) 241-268. | Zbl

[10] R.M. Dudley, Asymptotics of some relativistic Markov processes. Proc. Natl. Acad. Sci. USA 70 (1973) 3551-3555. | Zbl

[11] C. Frances, Géométrie et dynamique Lorentzienne conformes. École Normale Supérieure de Lyon (2002).

[12] R. Geroch, E.H. Kronheimer, and Roger Penrose, Ideal points in space-time. Proc. Roy. Soc. Lond. Ser. A 327 (1972) 545-567. | Zbl

[13] A. Grigor'Yan, 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. | Zbl

[14] 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).

[15] T.R. Hurd, The projective geometry of simple cosmological models. Proc. Roy. Soc. Lond. Ser. A 397 (1985) 233-243. | Zbl

[16] 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). | Zbl

[17] F.I. Karpelevič, V.N. Tutubalin and M.G. Šur, Limit theorems for compositions of distributions in the Lobačevskiĭ plane and space. Teor. Veroyatnost. i Primenen. 4 (1959) 432-436. | Zbl

[18] M. Liao, Lévy processes in Lie groups, volume 162 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2004). | Zbl

[19] J. Neveu, Mathematical foundations of the calculus of probability. Translated by Amiel Feinstein. Holden-Day Inc., San Francisco, Californie (1965). | Zbl

[20] 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). | Zbl

[21] R.G. Pinsky, Positive harmonic functions and diffusion, volume 45 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995). | Zbl

[22] 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. | Zbl

[23] A. Raugi, Fonctions harmoniques sur les groupes localement compacts à base dénombrable. Bull. Soc. Math. France, Mémoire 54 (1977) 5-118. | Numdam | Zbl

[24] A. Raugi, Périodes des fonctions harmoniques bornées. In Seminar on Probability, Rennes, 1978 (French). Exposé No. 10. Univ. Rennes, France (1978).

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