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Regenerative processes for Poisson zero polytopes

Part of: Ergodic theory Markov processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  29 November 2018

Servet Martínez  and
Werner Nagel
Servet Martínez*
Affiliation:
Universidad de Chile
Werner Nagel*
Affiliation:
Friedrich-Schiller-Universität Jena
*
* Postal address: Departamento Ingeniería Matemática and Centro Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile. Email address: smartine@dim.uchile.cl
** Postal address: Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany. Email address: werner.nagel@uni-jena.de
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Abstract

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Let (Mt:t>0) be a Markov process of tessellations of ℝℓ, and let (𝒞t:t>0) be the process of their zero cells (zero polytopes), which has the same distribution as the corresponding process for Poisson hyperplane tessellations. In the present paper we describe the stationary zero cell process (at𝒞at:t∈ℝ),a>1, in terms of some regenerative structure and we show that it is a Bernoulli flow. An important application is to STIT tessellation processes.

Keywords

Stochastic geometryrandom tessellationPoisson hyperplane tessellationSTIT tessellationzero polytopeBernoulli flowregenerative process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 37A25: Ergodicity, mixing, rates of mixing 37A35: Entropy and other invariants, isomorphism, classification

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1217 - 1226
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Asmussen, S. (2003).Applied Probability and Queues,2nd edn.Springer,New York.Google Scholar
[2]Ethier, S. N. and Kurtz, T. G. (1986).Markov Processes.John Wiley,New York.Google Scholar
[3]Grimmett, G. R. and Stirzaker, D. R. (1992).Probability and Random Processes: Problems and Solutions.Oxford University Press.Google Scholar
[4]Lindvall, T. (1992).Lectures on the Coupling Method.John Wiley,New York.Google Scholar
[5]Martínez, S. (2014).STIT tessellations are Bernoulli and standard.Ergodic Theory Dynam. Systems 34,876–892.Google Scholar
[6]Martínez, S. and Nagel, W. (2012).Ergodic Description of STIT tessellations.Stochastics 84,113–134.Google Scholar
[7]Mecke, J.,Nagel, W. and Weiss, V. (2008). A global construction of homogeneous random planar tessellations that are stable under iteration.Stochastics 80,51–67.Google Scholar
[8]Molchanov, I. (2005).Theory of Random Sets.Springer,London.Google Scholar
[9]Nagel, W. and Weiss, V. (2003).Limits of sequences of stationary planar tessellations.Adv. Appl. Prob. 35,123–138.Google Scholar
[10]Nagel, W. and Weiss, V. (2005).Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration.Adv. Appl. Prob. 37,859–883.Google Scholar
[11]Ornstein, D. S. (1974).Ergodic Theory, Randomness and Dynamical Systems.Yale University Press.Google Scholar
[12]Ross, S. M. (1983).Stochastic Processes.John Wiley,New York.Google Scholar
[13]Schneider, R. and Weil, W. (2008).Stochastic and Integral Geometry.Springer,Berlin.Google Scholar