Poisson因子分裂
Poisson splitting by factors
摘要：
给定${\mathbb{R}}^d$上的一个具有强度$λ$的齐次泊松过程，我们证明了可以将点划分为两个集合，作为该过程的确定性函数，并且以等距等变的方式，使得每个集合的点形成一个齐次泊松过程，任何给定的强度对之和为$λ$ .特别是，这回答了一个问题球[电子。可能吧10（2005）60- 69]，他证明了在$d=1$中，泊松点可以类似地被划分（通过双等变函数），使得一个集合形成较低强度的泊松过程，并询问对于所有$d$是否可能相同。我们不知道是否有可能类似地添加点（再次选择作为一个确定性的功能的泊松过程），以获得一个泊松过程更高的强度，但我们证明，这是不可能的，在一个额外的finitariness条件。
英文摘要：
Given a homogeneous Poisson process on ${\mathbb{R}}^d$ with intensity $λ$, 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 $λ$. 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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