no. 13-14
Probability Theory
On some relativistic-covariant stochastic processes in Lorentzian space-times

Presented by: Marc Yor

Émery, Michel1
1 IRMA, 7, rue René Descartes, 67084 Strasbourg cedex, France
Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 817-820.

A Lorentz-covariant relativistic Brownian motion has been defined by Dudley in the framework of special relativity, and extended to general relativity by Franchi and Le Jan. It is a random timelike curve of class C1, the world-line of a particle with an intrinsic (i.e., relativistically covariant) law of motion. Building on the Franchi–Le Jan process, we propose a possible definition for random spacelike curves enjoying relativistic covariance; they are more regular (at least C2) than the timelike ones.

Défini en relativité restreinte par Dudley, le mouvement brownien relativiste à covariance lorentzienne a été étendu à la relativité générale par Franchi et Le Jan. C'est une courbe aléatoire de classe C1, de genre temps, et dont la dynamique jouit de la covariance relativiste. En utilisant le processus de Franchi et Le Jan, nous définissons des courbes aléatoires de genre espace et à dynamique covariante. Ces courbes sont plus régulières (C2 au moins) que celles de Franchi et Le Jan.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.04.015
Émery, Michel 1

1 IRMA, 7, rue René Descartes, 67084 Strasbourg cedex, France 
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Émery, Michel. On some relativistic-covariant stochastic processes in Lorentzian space-times. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 817-820. doi : 10.1016/j.crma.2009.04.015. http://www.numdam.org/articles/10.1016/j.crma.2009.04.015/

[1] Dudley, R.M. A note on Lorentz-invariant Markov processes in relativistic phase-space, Ark. Math., Volume 6 (1966), pp. 241-268

[2] Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys., Volume 17 (1905), pp. 549-560

[3] Franchi, J.; Le Jan, Y. Relativistic diffusions and Schwarzschild geometry, Comm. Pure Appl. Math., Volume 60 (2007), pp. 187-251

[4] O'Neill, B. Semi-Riemannian Geometry, Academic Press, New York, 1983

[5] G. Schay, The equations of diffusion in the special theory of relativity, Ph.D. Thesis, Princeton University, 1961

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