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Ergodic Poisson splittings. (English) Zbl 07226360
Summary: 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.
##### MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G57 Random measures 37A50 Dynamical systems and their relations with probability theory and stochastic processes 37A40 Nonsingular (and infinite-measure preserving) transformations
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