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

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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

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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

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