A stochastic approach to relativistic diffusions

Bailleul, Ismaël

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 760-795.

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.

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.

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