Invariance of Poisson measures under random transformations
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, p. 947-972

We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai (Probab. Theory Related Fields 103 (1995) 409-429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure.

Nous montrons que les mesures de Poisson sont invariantes par les transformations aléatoires qui préservent les mesures d'intensité, et dont le gradient aux différences finies satisfait une condition d'annulation cyclique. La preuve de ce résultat repose sur des identités de moments d'intérêt indépendant pour les intégrales stochastiques de Poisson adaptées et anticipantes, et est inspirée par la méthode de Üstünel et Zakai (Probab. Theory Related Fields 103 (1995) 409-429) sur l'espace de Wiener, bien que l'algèbre correspondante soit plus compliquée que dans le cas Wiener. Les exemples d'application incluent des transformations conditionnées par des ensembles aléatoires tels que l'enveloppe convexe d'une mesure aléatoire de Poisson.

DOI : https://doi.org/10.1214/11-AIHP422
Classification:  60G57,  60G30,  60G55,  60H07,  28D05,  28C20,  37A05
Keywords: Poisson measures, random transformations, invariance, Skorohod integral, moment identities
@article{AIHPB_2012__48_4_947_0,
     author = {Privault, Nicolas},
     title = {Invariance of Poisson measures under random transformations},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {4},
     year = {2012},
     pages = {947-972},
     doi = {10.1214/11-AIHP422},
     zbl = {1278.60084},
     mrnumber = {3052400},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_4_947_0}
}
Privault, Nicolas. Invariance of Poisson measures under random transformations. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, pp. 947-972. doi : 10.1214/11-AIHP422. http://www.numdam.org/item/AIHPB_2012__48_4_947_0/

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