Shape-Driven Nested Markov Tessellations

TL;DR

This paper introduces a broad class of stationary random tessellations called shape-driven nested Markov tessellations, constructed via a recursive split process governed by Markovian kernels, with results on existence, uniqueness, and geometric properties.

Contribution

It provides the first explicit global construction and proves existence and uniqueness of shape-driven nested Markov tessellations, extending classical iteration stable tessellations.

Findings

Existence and uniqueness of the tessellations under certain conditions.

Explicit construction of the tessellations.

Analysis of typical cell and geometric properties.

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.