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Annales scientifiques de l'ENS - Titles - série 4, 50 (2017)

Titles < série 4, 50

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 50, fascicule 6 (2017)

Élise Janvresse, Emmanuel Roy, Thierry de la Rue
Poisson suspensions and SuShis
Annales scientifiques de l'ENS 50, fascicule 6 (2017), 1301-1334

Télécharger cet article : Fichier PDF

Résumé :
Suspensions de Poisson et SuShis
Dans cet article, nous démontrons qu'un processus ponctuel ergodique avec des moments de tous ordres, dirigé par une transformation préservant une mesure infinie qui vérifie certaines propriétés, est nécessairement une superposition de processus de Poisson décalés. Ce résultat de rigidité a de nombreuses implications en termes de couplages et de disjonction pour la suspension de Poisson associée. En particulier, nous démontrons que ses autocouplages ergodiques sont des couplages poissoniens, obtenant ainsi un analogue, dans le contexte poissonien, de la propriété GAG des systèmes dynamiques gaussiens.

Mots-clefs : Processus ponctuel, suspension de Poisson, couplages.

Abstract:
In this paper, we prove that ergodic point processes with moments of all orders, driven by particular infinite measure preserving transformations, have to be a superposition of shifted Poisson processes. This rigidity result has a lot of implications in terms of joining and disjointness for the corresponding Poisson suspension. In particular, we prove that its ergodic self-joinings are Poisson joinings, which provides an analog, in the Poissonian context, of the GAG property for Gaussian dynamical systems.

Keywords: Point process, Poisson suspension, joinings.

Class. math. : 37A50, 60G55, 37A40


ISSN : 0012-9593
DOI : doi:10.24033/asens.2346
Publié avec le concours de : Centre National de la Recherche Scientifique

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