Source: Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 3
(2010), 760-795.
A new class of relativistic diffusions encompassing all the previously studied examples has recently been introduced in the article of C. Chevalier and F. Debbasch (J. Math. Phys. 49 (2008) 043303), both in a heuristic and analytic way. A stochastic approach of these processes is proposed here, in the general framework of Lorentzian geometry. In considering the dynamics of the random motion in strongly causal spacetimes, we are able to give a simple definition of the one-particle distribution function associated with each process of the class and prove its fundamental property. This result not only provides a dynamical justification of the analytical approach developped up to now (enabling us to recover many of the results obtained so far), but it provides a new general H-theorem. It also sheds some light on the importance of the large scale structure of the manifold in the asymptotic behaviour of the Franchi–Le Jan process. This approach is also the source of many interesting questions that have no analytical counterparts.
C. Chevalier et F. Debbasch ont récemment introduit dans l’article (J. Math. Phys. 49 (2008) 043303) une nouvelle classe de diffusions relativistes comprenant toutes celles étudiées jusqu’̀à présent. Leur approche est heuristique et analytique. On propose dans cet article une approche stochastique de cette classe de processus, dans le cadre général d’une variété lorentzienne quelconque. Le cas des variétés fortement causales permet de donner une définition claire et simple de la “one-particle distribution function” associée ̀à chacun de ces processus et donne un cadre adéquat pour y prouver une propriété fondamentale. Ce résultat donne non seulement une justification dynamique de l’approche anaytique utilisée jusqu’̀à présent (recouvrant au passage la plupart des résultats obtenus jusqu’alors), mais il fournit aussi un H-théorème général. Il met aussi en lumière l’importance de la structure ̀à grande échelle de la variété dans le comportement asymptotique de la diffusion de Franchi–Le Jan. Cette approche est aussi la source de nombreuses questions intéressantes qui n’ont pas leur pendant analytique.
References
[1] C. Ane, S. Blachère, D. Chafaï, P. Fougère, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer. Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses [Panoramas and Syntheses] 10. Société Mathématique de France, Paris, 2000. With a preface by Dominique Bakry and Michel Ledoux.
[2] M. T. Anderson. The Dirichlet problem at infinity for manifolds of negative curvature. J. Differential Geom. 18 (1984) 701–721.
Mathematical Reviews (MathSciNet):
MR730923
[3] N. Andersson and G. L. Comer. Relativistic fluid dynamics: Physics for many different scales. Living Rev. Relativity 10 (2007) 1.
[4] J. Angst and J. Franchi. Central limit theorem for a class of relativistic diffusions. J. Math. Phys. 48 (2007) 083101.
[5] I. Bailleul. Poisson boundary of a relativistic diffusion. Probab. Theory Related Fields 141 (2008) 283–330.
[6] I. Bailleul and A. Raugi. Where does randomness lead in spacetime?
ESAIM Probab. Stat. 13 (2008) DOI:
10.1051/ps:2008021.
[7] C. Barbachoux, F. Debbasch and J. P. Rivet. Covariant kolmogorov equation and entropy current for the relativistic Ornstein–Uhlenbeck process. European J. Phys. B 23 (2001) 487–496.
[8] C. Barbachoux, F. Debbasch and J. P. Rivet. The spatially one-dimensinal relativistic Ornstein–Uhlenbeck process in an arbitrary inertial frame. European J. Phys. 19 (2001) 37–47.
[9] C. Chevalier and F. Debbasch. Relativistic diffusions: A unifying approach. J. Math. Phys. 49 (2008) 043303.
[10] C. Chevalier and F. Debbasch. A unifying approach to relativistic diffusions and H-theorems. Modern Phys. Lett. B 22 (2008) 383–392.
[11] T. M. Cover and J. A. Thomas. Elements of Information Theory, 2nd edition. Wiley, Hoboken, NJ, 2006.
[12] F. Debbasch. A diffusion process in curved space–time. J. Math. Phys. 45 (2004) 2744–2760.
[13] F. Debbasch, K. Mallick and J. P. Rivet. Relativistic Ornstein–Uhlenbeck process. J. Statist. Phys. 88 (1997) 945–966.
[14] F. Debbasch, J. P. Rivet and W. A. van Leeuwen. Invariance of the relativistic one-particle distribution function. Physica A 301 (2001) 181–195.
[15] F. Dowker, J. Henson and R. Sorkin. Quantum gravity phenomenology, Lorentz invariance and discreteness. Modern Phys. Lett. A 19 (2004) 1829–1840.
[16] R. M. Dudley. Lorentz-invariant Markov processes in relativistic phase space. Ark. Mat. 6 (1966) 241–268.
Mathematical Reviews (MathSciNet):
MR198540
[17] J. Dunkel and P. Hänggi. Theory of relativistic Brownian motion: The (1+3)-dimensional case. Phys. Rev. E (3) 72 (2005) 036106.
[18] E. B. Dynkin. Diffusions, Superdiffusions and Partial Differential Equations. American Mathematical Society Colloquium Publications 50. Amer. Math. Soc., Providence, RI, 2002.
[19] E. B. Dynkin and A. A. Yushkevich. Controlled Markov Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 235. Springer, Berlin, 1979. Translated from the Russian original by J. M. Danskin and C. Holland.
[20] J. Franchi. Relativistic diffusion in gödel’s universe. Commun. Math. Phys. 290 (2009) 523–555.
[21] J. Franchi and Y. Le Jan. Relativistic diffusions and Schwarzschild geometry. Comm. Pure Appl. Math. 60 (2007) 187–251.
[22] 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.
[23] S. W. Hawking and R. Penrose. The singularities of gravitational collapse and cosmology. Proc. Roy. Soc. London Ser. A 314 (1970) 529–548.
Mathematical Reviews (MathSciNet):
MR264959
[24] W. Israel. The relativistic Boltzmann equation. In General Relativity (Papers in Honour of J. L. Synge) 201–241. Clarendon Press, Oxford, 1972.
Mathematical Reviews (MathSciNet):
MR503418
[25] F. Juttner. Die relativistische quantentheorie des idealen gases. Zeitschr. Phys. 47 (1928) 542–566.
[26] Y. Kifer. Brownian motion and positive harmonic functions on complete manifolds of nonpositive curvature. In From Local Times to Global Geometry, Control and Physics (Coventry, 1984/85). Pitman Res. Notes Math. Ser. 150 187–232. Longman, Harlow, 1986.
Mathematical Reviews (MathSciNet):
MR894531
[27] P. Malliavin. Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 313. Springer, Berlin, 1997.
[28] L. Markus. Global Lorentz geometry and relativistic Brownian motion. In From Local Times to Global Geometry, Control and Physics (Coventry, 1984/85). Pitman Res. Notes Math. Ser. 150 273–286. Longman, Harlow, 1986.
Mathematical Reviews (MathSciNet):
MR894533
[29] R. S. Martin. Minimal positive harmonic functions. Trans. Amer. Math. Soc. 49 (1941) 137–172.
Mathematical Reviews (MathSciNet):
MR3919
[30] B. O’Neill. Semi-Riemannian Geometry. Pure and Applied Mathematics 103. Academic Press, New York, 1983. With applications to relativity.
Mathematical Reviews (MathSciNet):
MR719023
[31] R. G. Pinsky. A new approach to the Martin boundary via diffusions conditioned to hit a compact set. Ann. Probab. 21 (1993) 453–481.
[32] R. G. Pinsky. Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics 45. Cambridge Univ. Press, Cambridge, 1995.
[33] M. Rigotti and F. Debbasch. An H-theorem for the general relativistic Ornstein–Uhlenbeck process. J. Math. Phys. 46 (2005) 103303.
[34] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales 1. Cambridge Univ. Press, Cambridge, 2000. Foundations, reprint of the second (1994) edition.
