A stochastic approach to relativistic diffusions

Ismaël Bailleul

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 3, page 760-795
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

Bailleul, Ismaël. "A stochastic approach to relativistic diffusions." Annales de l'I.H.P. Probabilités et statistiques 46.3 (2010): 760-795. <http://eudml.org/doc/239289>.

@article{Bailleul2010,
abstract = {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.},
author = {Bailleul, Ismaël},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {diffusions; general relativity; harmonic functions; harmonic measure; -theorem},
language = {eng},
number = {3},
pages = {760-795},
publisher = {Gauthier-Villars},
title = {A stochastic approach to relativistic diffusions},
url = {http://eudml.org/doc/239289},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Bailleul, Ismaël
TI - A stochastic approach to relativistic diffusions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 3
SP - 760
EP - 795
AB - 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.
LA - eng
KW - diffusions; general relativity; harmonic functions; harmonic measure; -theorem
UR - http://eudml.org/doc/239289
ER -

References

top
  1. [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. Zbl0982.46026MR1845806
  2. [2] M. T. Anderson. The Dirichlet problem at infinity for manifolds of negative curvature. J. Differential Geom. 18 (1984) 701–721. Zbl0541.53036MR730923
  3. [3] N. Andersson and G. L. Comer. Relativistic fluid dynamics: Physics for many different scales. Living Rev. Relativity 10 (2007) 1. Zbl1255.85001
  4. [4] J. Angst and J. Franchi. Central limit theorem for a class of relativistic diffusions. J. Math. Phys. 48 (2007) 083101. Zbl1152.81316MR2349412
  5. [5] I. Bailleul. Poisson boundary of a relativistic diffusion. Probab. Theory Related Fields 141 (2008) 283–330. Zbl1138.60051MR2372972
  6. [6] I. Bailleul and A. Raugi. Where does randomness lead in spacetime? ESAIM Probab. Stat. 13 (2008) DOI: 10.1051/ps:2008021. Zbl1217.58021MR2640366
  7. [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. Zbl1230.82040
  8. [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. Zbl1230.82044
  9. [9] C. Chevalier and F. Debbasch. Relativistic diffusions: A unifying approach. J. Math. Phys. 49 (2008) 043303. Zbl1152.81373MR2412295
  10. [10] C. Chevalier and F. Debbasch. A unifying approach to relativistic diffusions and H-theorems. Modern Phys. Lett. B 22 (2008) 383–392. Zbl1151.82367MR2400933
  11. [11] T. M. Cover and J. A. Thomas. Elements of Information Theory, 2nd edition. Wiley, Hoboken, NJ, 2006. Zbl0762.94001MR2239987
  12. [12] F. Debbasch. A diffusion process in curved space–time. J. Math. Phys. 45 (2004) 2744–2760. Zbl1071.82031MR2067584
  13. [13] F. Debbasch, K. Mallick and J. P. Rivet. Relativistic Ornstein–Uhlenbeck process. J. Statist. Phys. 88 (1997) 945–966. Zbl0939.82015MR1467638
  14. [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. Zbl0978.82004
  15. [15] F. Dowker, J. Henson and R. Sorkin. Quantum gravity phenomenology, Lorentz invariance and discreteness. Modern Phys. Lett. A 19 (2004) 1829–1840. MR2079281
  16. [16] R. M. Dudley. Lorentz-invariant Markov processes in relativistic phase space. Ark. Mat. 6 (1966) 241–268. Zbl0171.39105MR198540
  17. [17] J. Dunkel and P. Hänggi. Theory of relativistic Brownian motion: The (1+3)-dimensional case. Phys. Rev. E (3) 72 (2005) 036106. MR2179917
  18. [18] E. B. Dynkin. Diffusions, Superdiffusions and Partial Differential Equations. American Mathematical Society Colloquium Publications 50. Amer. Math. Soc., Providence, RI, 2002. Zbl0999.60003MR1883198
  19. [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. Zbl0426.60063
  20. [20] J. Franchi. Relativistic diffusion in gödel’s universe. Commun. Math. Phys. 290 (2009) 523–555. Zbl1179.83021MR2525629
  21. [21] J. Franchi and Y. Le Jan. Relativistic diffusions and Schwarzschild geometry. Comm. Pure Appl. Math. 60 (2007) 187–251. Zbl1130.83006MR2275328
  22. [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. Zbl0927.58019MR1659871
  23. [23] S. W. Hawking and R. Penrose. The singularities of gravitational collapse and cosmology. Proc. Roy. Soc. London Ser. A 314 (1970) 529–548. Zbl0954.83012MR264959
  24. [24] W. Israel. The relativistic Boltzmann equation. In General Relativity (Papers in Honour of J. L. Synge) 201–241. Clarendon Press, Oxford, 1972. MR503418
  25. [25] F. Juttner. Die relativistische quantentheorie des idealen gases. Zeitschr. Phys. 47 (1928) 542–566. Zbl54.0987.01JFM54.0987.01
  26. [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. Zbl0611.31002MR894531
  27. [27] P. Malliavin. Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 313. Springer, Berlin, 1997. Zbl0878.60001MR1450093
  28. [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. Zbl0608.58046MR894533
  29. [29] R. S. Martin. Minimal positive harmonic functions. Trans. Amer. Math. Soc. 49 (1941) 137–172. Zbl67.0343.03MR3919JFM67.0343.03
  30. [30] B. O’Neill. Semi-Riemannian Geometry. Pure and Applied Mathematics 103. Academic Press, New York, 1983. With applications to relativity. Zbl0531.53051MR719023
  31. [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. Zbl0777.60075MR1207233
  32. [32] R. G. Pinsky. Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics 45. Cambridge Univ. Press, Cambridge, 1995. Zbl0858.31001MR1326606
  33. [33] M. Rigotti and F. Debbasch. An H-theorem for the general relativistic Ornstein–Uhlenbeck process. J. Math. Phys. 46 (2005) 103303. Zbl1111.82052MR2178600
  34. [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. Zbl0949.60003MR1796539

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.