## Electronic Journal of Probability

### Point-shift foliation of a point process

#### Abstract

A point-shift $F$ maps each point of a point process $\Phi$ to some point of $\Phi$. For all translation invariant point-shifts $F$, the $F$-foliation of $\Phi$ is a partition of the support of $\Phi$ which is the discrete analogue of the stable manifold of $F$ on $\Phi$. It is first shown that foliations lead to a classification of the behavior of point-shifts on point processes. Both qualitative and quantitative properties of foliations are then established. It is shown that for all point-shifts $F$, there exists a point-shift $F_\bot$, the orbits of which are the $F$-foils of $\Phi$, and which is measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one another.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 19, 25 pp.

Dates
Accepted: 4 November 2017
First available in Project Euclid: 23 February 2018

https://projecteuclid.org/euclid.ejp/1519354948

Digital Object Identifier
doi:10.1214/17-EJP123

Mathematical Reviews number (MathSciNet)
MR3771756

Zentralblatt MATH identifier
06868364

#### Citation

Baccelli, Francois; Haji-Mirsadeghi, Mir-Omid. Point-shift foliation of a point process. Electron. J. Probab. 23 (2018), paper no. 19, 25 pp. doi:10.1214/17-EJP123. https://projecteuclid.org/euclid.ejp/1519354948

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