Poisson splitting by factors

TL;DR

This paper demonstrates that a homogeneous Poisson process in any dimension can be deterministically partitioned into two independent Poisson processes with specified intensities summing to the original, answering a longstanding question.

Contribution

It proves the existence of an isometry-equivariant deterministic partition of Poisson processes into two Poisson processes with arbitrary combined intensities, extending previous 1D results.

Findings

Partition exists for any dimension and intensities

Partition is isometry-equivariant and deterministic

Adding points to form higher intensity Poisson processes is impossible under certain conditions

Abstract

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.