Poisson splitting by factors

Details Poisson splitting by factors Poisson splitting by factors

2009-08-24

arxiv.org/abs/0908.3409v3

Annals of Probability 2011, Vol. 39, No. 5, 1938-1982

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/11-AOP651 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/11-AOP651

Given a homogeneous Poisson process on ${\mathbb{R}}^d$ with intensity $\lambda$, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to $\lambda$. In particular, this answers a question of Ball [Electron. Commun. Probab. 10 (2005) 60--69], who proved that in $d=1$, the Poisson points may be similarly partitioned (via a translation-equivariant function) so that one set forms a Poisson process of lower intensity, and asked whether the same is possible for all $d$. We do not know whether it is possible similarly to add points (again chosen as a deterministic function of a Poisson process) to obtain a Poisson process of higher intensity, but we prove that this is not possible under an additional finitariness condition.

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