Branching random tessellations with interaction - a last project of Tomasz Schreiber
Branching random tessellations (BRTs) are stochastic processes that transform any initial tessellation of into a finer tessellation by means of random cell divisions in continuous time. In a draft entrusted to me by Tomasz Schreiber shortly before his death, he started an investigation of the thermodynamics of these objects. A special case are the so-called STIT tessellations, for which all cells split up independently. By way of contrast, the cells of a BRT are allowed to interact, in that the division rule for each cell may depend on the structure and past of the surrounding tessellation. In addition, the cells are endowed with an internal property, called their colour. Under a suitable condition, the cell interaction of a BRT can be specified by a measure kernel that determines the division rules of all cells and gives rise to a Gibbsian characterisation of BRTs. For translation invariant BRTs, one can introduce an inner entropy density relative to a STIT tessellation. Together with an inner energy density for a given moderate division kernel, this leads to a variational principle for BRTs with this prescribed kernel, and further to an existence result for such BRTs.