A Proof of a Conjecture by Mecke for STIT tessellations

TL;DR

This paper proves Mecke's conjecture that a specific continuous-time model of STIT tessellations is equivalent to the original process, clarifying its behavior and relation to other models.

Contribution

It provides a proof of Mecke's conjecture, establishing the equivalence between the continuous-time model and the original STIT tessellation process.

Findings

Proof of Mecke's conjecture confirming the equivalence

Relation to Cowan's continuous-time model clarified

Enhanced understanding of Mecke's process in continuous time

Abstract

The STIT tessellation process was introduced and examined by Mecke, Nagel and Wei{\ss}; many of its main characteristics are contained in a paper published by Nagel and Wei{\ss} in 2005. In a paper published in 2010, Mecke introduced another process in discrete time. With a geometric distribution whose parameter depends on the time, he reaches a continuous-time model. In his Conjecture 3, he assumed this continuous-time model to be equivalent to STIT. In the present paper, that conjecture is proven. An interesting relation arises to a continuous-time version of the equally-likely model classified by Cowan in 2010. This will also clarify how Mecke's model works as a process in continuous time.