Advances in Applied Probability

Consistency of constructions for cell division processes

Werner Nagel and Eike Biehler

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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 WV 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.

Article information

Adv. in Appl. Probab., Volume 47, Number 3 (2015), 640-651.

First available in Project Euclid: 8 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60J75: Jump processes

Stochastic geometry random tessellation iteration/nesting of tessellations STIT tessellation spatial consistency


Nagel, Werner; Biehler, Eike. Consistency of constructions for cell division processes. Adv. in Appl. Probab. 47 (2015), no. 3, 640--651. doi:10.1239/aap/1444308875.

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  • Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and Its Applications, 3rd edn. John Wiley, Chichester.
  • Cowan, R. (2010). New classes of random tessellations arising from iterative division of cells. Adv. Appl. Prob. 42, 26–47.
  • Georgii, H.-O., Schreiber, T. and Thäle, C. (2015). Branching random tessellations with interaction: a thermodynamic view. Ann. Prob. 43, 1892–1943.
  • Klenke, A. (2014). Probability Theory, 2nd edn. Springer, London.
  • Martínez, S. and Nagel, W. (2012). Ergodic description of STIT tessellations. Stochastics 84, 113–134.
  • 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) 14pp.
  • 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.
  • Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
  • Schreiber, T. and Thäle, C. (2013). Shape-driven nested Markov tessellations. Stochastics 85, 510–531.