| Poisson Splitting by Factors (2009) | |||||||||
Abstract | |||||||||
| Given a homogeneous Poisson process on R^d with intensity L, 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 L. In particular, this answers a question of Ball, 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 was 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.. Comment: 55 pages, 3 figures | |||||||||
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