## The Annals of Probability

### Branching random tessellations with interaction: A thermodynamic view

#### Abstract

A branching random tessellation (BRT) is a stochastic process that transforms a coarse initial tessellation of $\mathbb{R}^{d}$ into a finer tessellation by means of random cell divisions in continuous time. This concept generalises the so-called STIT tessellations, for which all cells split up independently of each other. Here, we allow the cells to interact, in that the division rule for each cell may depend on the structure of the surrounding tessellation. Moreover, we consider coloured tessellations, for which each cell is marked with an internal property, called its colour. Under a suitable condition, the cell interaction of a BRT can be specified by a measure kernel, the so-called division kernel, that determines the division rules of all cells and gives rise to a Gibbsian characterisation of BRTs. For translation invariant BRTs, we introduce an “inner” entropy density relative to a STIT tessellation. Together with an inner energy density for a given “moderate” division kernel, this leads to a variational principle for BRTs with this prescribed kernel, and further to an existence result for such BRTs.

#### Article information

Source
Ann. Probab. Volume 43, Number 4 (2015), 1892-1943.

Dates
Revised: November 2013
First available in Project Euclid: 3 June 2015

https://projecteuclid.org/euclid.aop/1433341323

Digital Object Identifier
doi:10.1214/14-AOP923

Mathematical Reviews number (MathSciNet)
MR3353818

Zentralblatt MATH identifier
1320.60035

#### Citation

Georgii, Hans-Otto; Schreiber, Tomasz; Thäle, Christoph. Branching random tessellations with interaction: A thermodynamic view. Ann. Probab. 43 (2015), no. 4, 1892--1943. doi:10.1214/14-AOP923. https://projecteuclid.org/euclid.aop/1433341323

#### References

• [1] Arak, T. and Surgailis, D. (1989). Markov fields with polygonal realizations. Probab. Theory Related Fields 80 543–579.
• [2] Arak, T. and Surgailis, D. (1991). Consistent polygonal fields. Probab. Theory Related Fields 89 319–346.
• [3] Bertin, E., Billiot, J.-M. and Drouilhet, R. (1999). Existence of Delaunay pairwise Gibbs point process with superstable component. J. Stat. Phys. 95 719–744.
• [4] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Univ. Press, Cambridge.
• [5] Cattiaux, P., Roelly, S. and Zessin, H. (1996). Une approche gibbsienne des diffusions browniennes infini-dimensionnelles. Probab. Theory Related Fields 104 147–179.
• [6] Dai Pra, P. (1993). Large deviations and stationary measures for interacting particle systems. Stochastic Process. Appl. 48 9–30.
• [7] Dai Pra, P., Roelly, S. and Zessin, H. (2002). A Gibbs variational principle in space-time for infinite-dimensional diffusions. Probab. Theory Related Fields 122 289–315.
• [8] Dereudre, D., Drouilhet, R. and Georgii, H.-O. (2012). Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Related Fields 153 643–670.
• [9] Dereudre, D. and Georgii, H.-O. (2009). Variational characterisation of Gibbs measures with Delaunay triangle interaction. Electron. J. Probab. 14 2438–2462.
• [10] Deuschel, J.-D. (1986). Nonlinear smoothing of infinite-dimensional diffusion processes. Stochastics 19 237–261.
• [11] Feller, W. (1940). On the integro-differential equations of purely discontinuous Markoff processes. Trans. Amer. Math. Soc. 48 488–515.
• [12] Föllmer, H. and Snell, J. L. (1977). An “inner” variational principle for Markov fields on a graph. Z. Wahrsch. Verw. Gebiete 39 187–195.
• [13] Georgii, H.-O. (2011). Gibbs Measures and Phase Transitions, 2nd ed. de Gruyter, Berlin.
• [14] Israel, R. B. (1979). Convexity in the Theory of Lattice Gases. Princeton Univ. Press, Princeton, NJ.
• [15] Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie Verlag, Berlin.
• [16] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
• [17] 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.
• [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] Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, Chichester.
• [20] Redenbach, C. and Thäle, C. (2013). On the arrangement of cells in planar STIT and Poisson line tessellations. Methodol. Comput. Appl. Probab. 15 643–654.
• [21] Ross, S. M. (2003). Introduction to Probability Models, 8th ed. Academic Press, Amsterdam.
• [22] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
• [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. (2012). Second-order theory for iteration stable tessellations. Probab. Math. Statist. 32 281–300.
• [26] Schreiber, T. and Thäle, C. (2013). Shape-driven nested Markov tessellations. Stochastics 85 510–531.
• [27] Schreiber, T. and Thäle, C. (2013). Limit theorems for iteration stable tessellations. Ann. Probab. 41 2261–2278.
• [28] Schreiber, T. and Thäle, C. (2013). Geometry of iteration stable tessellations: Connection with Poisson hyperplanes. Bernoulli 19 1637–1654.
• [29] Stoyan, D., Kendall, D. G. and Mecke, J. (1995). Stochastic Geometry, 2nd ed. Wiley, Chichester.
• [30] Thäle, C., Weiss, V. and Nagel, W. (2012). Spatial STIT tessellations: Distributional results for I-segments. Adv. in Appl. Probab. 44 635–654.
• [31] Varadhan, S. R. S. (1988). Large deviations and applications. In École D’Été de Probabilités de Saint-Flour XV–XVII, 198587. Lecture Notes in Math. 1362 1–49. Springer, Berlin.