zbMATH — the first resource for mathematics

Point-shift foliation of a point process. (English) Zbl 1454.60061
Summary: 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.
Reviewer: Reviewer (Berlin)

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
60G10 Stationary stochastic processes
60G57 Random measures
Full Text: DOI Euclid
[1] D.V. Anosov, Foliations, Springer Verlag, 2001, in Encyclopedia of Mathematics.
[2] F. Baccelli, B. Blaszczyszyn, and M.-O. Haji-Mirsadeghi, Optimal paths on the space-time SINR random graph, Advances in Applied Probability 43 (2011), no. 1, 131–150. · Zbl 1223.60011
[3] F. Baccelli and P. Brémaud, Elements of queueing theory, Springer Verlag, 2003.
[4] F. Baccelli and M.-O. Haji-Mirsadeghi, Point-map-probabilities of a point process and Mecke’s invariant measure equation, Ann. Probability 45 (2017), no. 3, 1723–1751. · Zbl 1375.60091
[5] D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes. Vol. II, second ed., Springer, 2008. · Zbl 0657.60069
[6] P. A. Ferrari, C. Landim, and H. Thorisson, Poisson trees, succession lines and coalescing random walks, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 2, 141–152. · Zbl 1042.60064
[7] S. Gangopadhyay, R. Roy, and A. Sarkar, Random oriented trees : a model of drainage networks, Annals of Applied Probability 14 (2004), 1242–1266. · Zbl 1047.60098
[8] M. Heveling and G. Last, Characterization of Palm measures via bijective point-shifts, Ann. Probab. 33 (2005), no. 5, 1698–1715. · Zbl 1111.60029
[9] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, 1995. · Zbl 0878.58020
[10] D. Koenig, Theory of finite and infinite graphs, Birkhauser, 1990.
[11] G. Last and H. Thorisson, Invariant transports of stationary random measures and mass-stationarity, Ann. Probab. 37 (2009), no. 2, 790–813. · Zbl 1176.60036
[12] J. Mecke, Invarianzeigenschaften allgemeiner Palmscher Maße, Math. Nachr. 65 (1975), 335–344. · Zbl 0301.28014
[13] H. Thorisson, Coupling, stationarity, and regeneration, Springer-Verlag, 2000. · Zbl 0949.60007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.