Bernoulli

  • Bernoulli
  • Volume 19, Number 5A (2013), 1637-1654.

Geometry of iteration stable tessellations: Connection with Poisson hyperplanes

Tomasz Schreiber and Christoph Thäle

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Abstract

Since the seminal work by Nagel and Weiss, the iteration stable (STIT) tessellations have attracted considerable interest in stochastic geometry as a natural and flexible, yet analytically tractable model for hierarchical spatial cell-splitting and crack-formation processes. We provide in this paper a fundamental link between typical characteristics of STIT tessellations and those of suitable mixtures of Poisson hyperplane tessellations using martingale techniques and general theory of piecewise deterministic Markov processes (PDMPs). As applications, new mean values and new distributional results for the STIT model are obtained.

Article information

Source
Bernoulli Volume 19, Number 5A (2013), 1637-1654.

Dates
First available in Project Euclid: 5 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1383661197

Digital Object Identifier
doi:10.3150/12-BEJ424

Mathematical Reviews number (MathSciNet)
MR3129028

Zentralblatt MATH identifier
1291.60021

Keywords
infinite divisibility iteration/nesting Markov process martingale theory piecewise deterministic Markov process random tessellation stochastic geometry stochastic stability

Citation

Schreiber, Tomasz; Thäle, Christoph. Geometry of iteration stable tessellations: Connection with Poisson hyperplanes. Bernoulli 19 (2013), no. 5A, 1637--1654. doi:10.3150/12-BEJ424. https://projecteuclid.org/euclid.bj/1383661197.


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References

  • [1] Baumstark, V. and Last, G. (2009). Gamma distributions for stationary Poisson flat processes. Adv. in Appl. Probab. 41 911–939.
  • [2] Beil, M., Eckel, S., Fleischer, F., Schmidt, H., Schmidt, V. and Walther, P. (2006). Fitting of random tessellation models to keratin filament networks. J. Theoret. Biol. 241 62–72.
  • [3] Beil, M., Lück, S., Fleischer, F., Portet, S., Arendt, W. and Schmidt, V. (2009). Simulating the formation of keratin filament networks by a piecewise-deterministic Markov process. J. Theor. Biol. 256 518–532.
  • [4] Chiquet, J., Limnios, N. and Eid, M. (2009). Piecewise deterministic Markov processes applied to fatigue crack growth modelling. J. Statist. Plann. Inference 139 1657–1667.
  • [5] Gikhman, I.I. and Skorokhod, A.V. (1974). The Theory of Stochastic Processes. II. Berlin: Springer.
  • [6] Jacobsen, M. (2006). Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Probability and Its Applications. Boston, MA: Birkhäuser.
  • [7] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). New York: Springer.
  • [8] Karatzas, I. and Shreve, S.E. (1998). Brownian Motion and Stochastic Calculus, 2nd ed. New York: Springer.
  • [9] Kato, T. (1966). Perturbation Theory for Linear Operators. Berlin: Springer.
  • [10] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Berlin: Springer.
  • [11] Lautensack, C. (2008). Fitting three-dimensional Laguerre tessellations to foam structures. J. Appl. Stat. 35 985–995.
  • [12] Lautensack, C. and Sych, T. (2005). 3d image analysis of foams using random tessellations. In Proceedings of the 9th European Congress on Stereology and Image Analysis and 7th STERMAT International Conference on Stereology and Image Analysis in Materials Science in Zakopane 147–153.
  • [13] Mecke, J., Nagel, W. and Weiss, V. (2007). Length distributions of edges in planar stationary and isotropic STIT tessellations. J. Contemp. Math. Anal. 42 28–43.
  • [14] 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.
  • [15] Mecke, J., Nagel, W. and Weiss, V. (2008). The iteration of random tessellations and a construction of a homogeneous process of cell divisions. Adv. in Appl. Probab. 40 49–59.
  • [16] Nagel, W., Mecke, J., Ohser, J. and Weiss, V. (2008). A tessellation model for crack patterns on surfaces. Image Anal. Stereol. 27 73–78.
  • [17] Nagel, W. and Weiss, V. (2003). Limits of sequences of stationary planar tessellations. Adv. in Appl. Probab. 35 123–138. In honor of Joseph Mecke.
  • [18] Nagel, W. and Weiss, V. (2005). Crack STIT tessellations: Characterization of stationary random tessellations stable with respect to iteration. Adv. in Appl. Probab. 37 859–883.
  • [19] Nagel, W. and Weiss, V. (2008). Mean values for homogeneous STIT tessellations in 3D. Image Anal. Stereol. 27 29–37.
  • [20] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Probability and Its Applications (New York). Berlin: Springer.
  • [21] Schreiber, T. (2010). Limit theorems in stochastic geometry. In New Perspectives in Stochastic Geometry (W.S. Kendall and I. Molchanov, eds.). Oxford: Oxford Univ. Press.
  • [22] Schreiber, T. and Thäle, C. (2010). Typical geometry, second-order properties and central limit theory for iteration stable tessellations. Available at arXiv:1001.0990.
  • [23] Schreiber, T. and Thäle, C. (2010). Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane. Adv. in Appl. Probab. 42 913–935.
  • [24] Schreiber, T. and Thäle, C. (2011). Intrinsic volumes of the maximal polytope process in higher dimensional STIT tessellations. Stochastic Process. Appl. 121 989–1012.
  • [25] Schreiber, T. and Thäle, C. (2011). Second-order theory for iteration stable tessellations. Available at arXiv:1103.3959.
  • [26] Schreiber, T. and Thäle, C. (2013). Limit theorems for iteration stable tessellations. Ann. Probab. 41 2261–2278.
  • [27] Stoyan, D., Kendall, W.S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd ed. Chichester: Wiley.