Ergodic Poisson Splittings

Elise Janvresse (1) , Emmanuel Roy (2) , Thierry de La Rue (3)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
2 LAGA - Laboratoire Analyse, Géométrie et Applications
3 LMRS - Laboratoire de Mathématiques Raphaël Salem

In this paper we study splittings of a Poisson point process which are equivariant under a conservative transformation. We show that, if the Cartesian powers of this transformation are all ergodic, the only ergodic splitting is the obvious one, that is, a collection of independent Poisson processes. We apply this result to the case of a marked Poisson process: under the same hypothesis, the marks are necessarily independent of the point process and i.i.d. Under additional assumptions on the transformation, a further application is derived, giving a full description of the structure of a random measure invariant under the action of the transformation.

Domaines: Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Elise Janvresse, Emmanuel Roy, Thierry de La Rue. Ergodic Poisson Splittings. Annals of Probability, 2020, 48 (3), pp.1266-1285. ⟨10.1214/19-AOP1390⟩. ⟨hal-01912386⟩