Poisson suspensions and SuShis
[Suspensions de Poisson et SuShis]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 6, pp. 1301-1334.

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.

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.

Publié le :
DOI : 10.24033/asens.2346
Classification : 37A50, 60G55, 37A40
Keywords: Point process, Poisson suspension, joinings.
Mot clés : Processus ponctuel, suspension de Poisson, couplages. 
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     title = {Poisson suspensions and {SuShis}},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1301--1334},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
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Janvresse, Élise; Roy, Emmanuel; de la Rue, Thierry. Poisson suspensions and SuShis. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 6, pp. 1301-1334. doi : 10.24033/asens.2346. http://www.numdam.org/articles/10.24033/asens.2346/

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