A Mecke-type formula and Markov properties for STIT tessellation processes

TL;DR

This paper establishes a Mecke-type formula for STIT tessellation processes, deriving their Markov properties and analyzing the distribution of internal vertices in typical segments, advancing stochastic geometry understanding.

Contribution

It introduces a Mecke-type formula for space-time STIT tessellations and uses it to derive Markov properties and vertex distribution results.

Findings

Established a Mecke-type formula for STIT tessellations.

Derived Markov properties for associated processes.

Determined the distribution of internal vertices in typical segments.

Abstract

An analogue of the classical Mecke formula for Poisson point processes is proved for the class of space-time STIT tessellation processes. From this key identity the Markov property of a class of associated random processes is derived. This in turn is used to determine the distribution of the number of internal vertices of the typical maximal tessellation segment.