Ergodic Poisson Splittings
Probability
2018-11-21 v1 Dynamical Systems
Abstract
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.
Cite
@article{arxiv.1811.08174,
title = {Ergodic Poisson Splittings},
author = {Elise Janvresse and Emmanuel Roy and Thierry De La Rue},
journal= {arXiv preprint arXiv:1811.08174},
year = {2018}
}