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Regenerative processes for Poisson zero polytopes

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

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

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

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