A global construction of homogeneous random planar tessellations that are stable under iteration

Shape-driven nested Markov tessellations

“…As already mentioned before, the crucial feature of Theorem 1 is that the unique wholespace tessellation Y Φ (t) gets constructed in a global construction greatly reminiscent of the (by now) well known Mecke-Nagel-Weiss (MNW) global construction developed in [11] for the STIT processes. Hence, the connection between the incremental MNW-construction, represented here by the [Recursive split dynamics], and the corresponding global MNWconstruction, represented here by the global construction in our Subsection 5.1, carries over to the general class of shape-driven tessellations.…”

“…We close this paragraph by noting that the cells S * i eventually cover the whole space, which is ∞ i=0 S * i = R d . This can be seen as follows, see also [11,Lem. 4.1]: Denote by σ < t the largest birth time of a cell from the spinal growth process hitting the convex hull of B 1 (p) and the center c(S seed ) of the seed cell, where B 1 (p) is the ball around p ∈ R d with radius 1.…”

“…As already signaled above, our proof of the principal Theorem 1 goes by providing first a global construction for the infinite volume Φ-splitting tessellation, constituting a direct and natural extension of the procedure given by Mecke, Nagel and Weiss in [11,12] for STIT tessellations. We then show that the so-obtained tessellation is the unique one compatible with Φ in the full space R d and we verify a number of properties thereof, obtained as by-products of the argument.…”

“…An iteration stable random tessellation Y Λ (t) with driving measure Λ and time parameter t > 0 is constructed according to the following rules, originally due to Mecke, Nagel and Weiss, see [11,12,14], and hence referred to as the MNW-construction in the sequel. We begin at the time 0 with a compact and convex subset W ⊂ R d (called window here) having nonempty interior, regard it as the initial cell for the tessellation under construction and assign to it an exponentially distributed random lifetime with parameter Λ ( …”

“…[17,23]). Stationary random tessellations that are stable under the operation of iteration -so-called STIT tessellations -were introduced recently by Mecke, Nagel and Weiß [11,12,14] and they may serve as a new mathematical reference model beside the classical Poisson hyperplane or PoissonVoronoi tessellations. On the other hand, the first-order geometry of these tessellations is already determined by two parameters, the surface density and the so-called directional distribution.…”