A Markov process associated with plot-size distribution in Czech Land Registry and its number-theoretic properties

The size distribution of land plots is a result of land allocation processes in the past. In the absence of regulation this is a Markov process leading an equilibrium described by a probabilistic equation used commonly in the insurance and financial mathematics. We support this claim by analyzing the distribution of two plot types, garden and build-up areas, in the Czech Land Registry pointing out the coincidence with the distribution of prime number factors described by Dickman function in the first case. The distribution of commodities is an important research topic in economy – see [CC07] for an extensive literature overview. In this letter we focus on a particular case, the allocation of land representing a non-consumable commodity, and a way in which the distribution is reached. Generally speaking, it results from a process of random commodity exchanges between agents in the situation when the aggregate commodity volume is conserved, in other words, one deals with pure trading which leads commodity redistribution. Models of this type were recently intensively discussed [SGG06] and are usually referred to as kinetic exchange models. Our approach here will be different being based on the concept known as perpetuity. The latter is a random variable D that satisfies a stochastic fixed-point equation the form