Etude infinitésimale et asymptotique de certains flots stochastiques relativistes

Abstract : We study some Lévy processes with values in the isometry group of Minkowski, De Sitter and Anti-de-Sitter space-times. The isometry group is seen as the frame bundle of the space-time and the Lévy processes we consider are some lift of relativistic markovian processes with values in the unitary tangent bundle of the space-time. Theses processes are relativistic in the sense that theirs trajectories are time-like and their generators are invariant by the isometries of the space-time. In the first part of this work we adapt to the case of a general hypoelliptic diffusion a result of Ben Arous and Gradinaru concerning the singularity of the hypoelliptic Green function. We deduce of this a local Wiener criterion for the relativistic diffusion in the isometry group of Minkowski space-time. In the two last parts we are interested to the asymptotic behavior of the stochastic flow associated to these Lévy processes in the different considered space-times. Under a integrability condition on the Lévy measure we compute explicitly the Lyapunov coefficient for such flows in the isometry group of Minkowski space-time. Then, we do a similar work in the context of de Sitter and Anti-de-Sitter space-times limiting ourselves to the case of diffusions. In fine, we explicit the Poisson boundary of the diffusion in the isometry group of de Sitter space-time.
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https://tel.archives-ouvertes.fr/tel-00703181
Contributor : Camille Tardif <>
Submitted on : Friday, June 1, 2012 - 10:11:25 AM
Last modification on : Wednesday, May 16, 2018 - 5:22:03 PM
Long-term archiving on : Sunday, September 2, 2012 - 2:22:10 AM

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  • HAL Id : tel-00703181, version 1

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Camille Tardif. Etude infinitésimale et asymptotique de certains flots stochastiques relativistes. Probabilités [math.PR]. Université de Strasbourg, 2012. Français. ⟨tel-00703181v1⟩

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