ERGODIC POISSON SPLITTINGS

ERGODIC POISSON SPLITTINGS

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...

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Veröffentlicht in:The Annals of probability 2020-05, Vol.48 (3), p.1266-1285
Hauptverfasser: Janvresse, Élise, Roy, Emmanuel, de la Rue, Thierry
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Dynamical Systems
Mathematics
Probability
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Roy, Emmanuel
de la Rue, Thierry
description 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.
doi_str_mv 10.1214/19-AOP1390
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subjects Dynamical Systems
Mathematics
Probability
title ERGODIC POISSON SPLITTINGS
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