Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-29T05:40:42.455Z Has data issue: false hasContentIssue false
  • English
  • Français

Consistency of constructions for cell division processes

Part of: Geometric probability and stochastic geometry Markov processes

Published online by Cambridge University Press:  21 March 2016

Werner Nagel  and
Eike Biehler
Werner Nagel*
Affiliation:
Friedrich-Schiller-Universität Jena
Eike Biehler*
Affiliation:
Friedrich-Schiller-Universität Jena
*
∗ Postal address: Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, D-07737 Jena, Germany.
∗ Postal address: Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, D-07737 Jena, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a class of cell division processes in the Euclidean space ℝd, spatial consistency is investigated. This addresses the problem whether the distribution of the generated structures, restricted to a bounded set V, depends on the choice of a larger region W ⊃ V where the construction of the cell division process is performed. This can also be understood as the problem of boundary effects in the cell division procedure. It is known that the STIT tessellations are spatially consistent. In the present paper it is shown that, within a reasonable wide class of cell division processes, the STIT tessellations are the only ones that are consistent.

Keywords

Stochastic geometryrandom tessellationiteration/nesting of tessellationsSTIT tessellationspatial consistency

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60J75: Jump processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 640 - 651
Copyright
Copyright © Applied Probability Trust 2015 

References

Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and Its Applications, 3rd edn. John Wiley, Chichester.CrossRefGoogle Scholar
Cowan, R. (2010). New classes of random tessellations arising from iterative division of cells. Adv. Appl. Prob. 42, 26–47.CrossRefGoogle Scholar
Georgii, H.-O., Schreiber, T. and Thäle, C. (2015). Branching random tessellations with interaction: a thermodynamic view. Ann. Prob. 43, 1892–1943.CrossRefGoogle Scholar
Klenke, A. (2014). Probability Theory, 2nd edn. Springer, London.CrossRefGoogle Scholar
Martínez, S. and Nagel, W. (2012). Ergodic description of STIT tessellations. Stochastics 84, 113–134.Google Scholar
Mosser, L. J. and Matthäi, S. K. (2014). Tessellations stable under iteration. Evaluation of application as an improved stochastic discrete fracture modeling algorithm. International Discrete Fracture Network Engineering Conference (Vancouver, 2014) 14 pp.Google Scholar
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.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Schreiber, T. and Thäle, C. (2013). Shape-driven nested Markov tessellations. Stochastics 85, 510–531.Google Scholar