Regenerative processes for Poisson zero polytopes

TL;DR

This paper studies the stationary zero cell process in Poisson hyperplane tessellations, revealing its regenerative structure and Bernoulli flow properties, with applications to STIT tessellation processes.

Contribution

It introduces a regenerative structure for the stationary zero cell process and proves it is a Bernoulli flow, advancing understanding of Poisson tessellations.

Findings

The zero cell process has a regenerative structure.

The process is shown to be a Bernoulli flow.

Applications to STIT tessellations are demonstrated.

Abstract

Let $(M_t: t > 0)$ be a Markov process of tessellations of ${\mathbb R}^\ell$ and $({\cal C}_t:\, t > 0)$ the process of their zero cells (zero polytopes) which has the same distribution as the corresponding process for Poisson hyperplane tessellations. Let $a>1$. Here we describe the stationary zero cell process $(a^t {\cal C}_{a^t}:\, t\in {\mathbb R})$ in terms of some regenerative structure and we prove that it is a Bernoulli flow. An important application are the STIT tessellation processes.