Electronic Journal of Probability
- Electron. J. Probab.
- Volume 23 (2018), paper no. 19, 25 pp.
Point-shift foliation of a point process
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.
Electron. J. Probab., Volume 23 (2018), paper no. 19, 25 pp.
Received: 30 October 2016
Accepted: 4 November 2017
First available in Project Euclid: 23 February 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx] 60G10: Stationary processes 60G55: Point processes 60G57: Random measures
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