Shape-driven nested Markov tessellations

Abstract

A new and rather broad class of stationary (i.e. stochastically translation invariant) random tessellations of the d-dimensional Euclidean space is introduced, which are called shape-driven nested Markov tessellations. Locally, these tessellations are constructed by means of a spatio-temporal random recursive split dynamics governed by a family of Markovian split kernel, generalizing thereby the -by now classical -construction of iteration stable random tessellations. By providing an explicit global construction of the tessellations, it is shown that under suitable assumptions on the split kernels (shape-driven), there exists a unique time-consistent whole-space tessellation-valued Markov process of stationary random tessellations compatible with the given split kernels. Beside the existence and uniqueness result, the typical cell and some aspects of the first-order geometry of these tessellations are in the focus of our discussion.

Consistency of constructions for cell division processes

“…of the polytopes which constitute a tessellation. This confirms a conjecture in [9], where a rather general approach to cell division processes is treated. Usually, these tessellations are constructed in a bounded window W ⊂ R d .…”

“…In [3] and [9] the approach to cell division processes is considerably more general, in particular allowing for dependencies of the lifetimes and distributions of the dividing hyperplanes on the whole tessellation. But there, the problem of consistency is not addressed, except in the conjecture in [9] saying that only STIT tessellations and some related constructions are consistent.…”

Consistency of constructions for cell division processes

“…With a further restriction to homogeneity (spatial stationarity) and continuous time, we show that in this class the STIT (stable with respect to iteration) tessellations (introduced in [7]) are the only ones which are consistent. This confirms a conjecture in [9], where a rather general approach to cell division processes is treated.…”

Branching random tessellations with interaction: A thermodynamic view

“…[13], Section 8.3), we will show in Proposition 4.17 below that uniqueness does hold for suitable division kernels of bounded range in one spatial dimension. Uniqueness is also known in the noninteracting case (2.21) when the initial tessellation is degenerate and the density ϕ exhibits some regularity properties [26]. We leave it to the future to find sufficient conditions for uniqueness in higher dimensions as well as examples of bounded-range division kernels exhibiting phase transition.…”

“…These tessellations are constructed by means of a temporal random process of cell division, and thus live in space-time. They have attracted considerable interest because of its analytical tractability; see, for example, [20,[23][24][25][26][27][28] or [30].Our objects of study here generalise the STIT models in two respects. On the one hand, we consider coloured tessellations, for which each cell is equipped with an individual colour.…”