A Proof of a Conjecture by Mecke for STIT tessellations

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.