Geometry of iteration stable tessellations: Connection with Poisson hyperplanes

TL;DR

This paper establishes a fundamental connection between iteration stable tessellations and Poisson hyperplane tessellations, providing new analytical tools and results for understanding hierarchical spatial cell-splitting processes.

Contribution

It introduces a link between STIT tessellations and Poisson hyperplane tessellations using martingale techniques and PDMP theory, leading to new mean and distributional results.

Findings

Derived new mean values for STIT tessellations

Obtained distributional results for hierarchical cell-splitting

Connected STIT models with Poisson hyperplane processes

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.