PUB - Publikationen an der Universität Bielefeld

We consider a fragmentation process that describes a specific way of successively removing objects from a linear arrangement. The process is a Markov chain, which is formulated in both a continuous-time and a discrete-time version. We aim at the law of the process over time. To this end, we investigate sets of realisations of this process that agree with respect to a specific order of events and represent each such set by a rooted (binary) tree. The probability distribution of the fragmentation process is then given as the sum over all relevant tree probabilities. The probability of each tree is, in turn, obtained by M\"obius inversion on a suitable poset of all rooted forests that can be obtained from the tree by edge deletion; we call this poset the pruning poset. In continuous time, explicit expressions for the probabilities of the trees transpire without much effort. In discrete time, however, dependencies within the fragments, which get lost over time, make it difficult to obtain analogous explicit results. We therefore construct an auxiliary process for every given tree, which is i.i.d. over time, and which allows to give a pathwise construction of realisations that match the tree. The fragmentation process arises in population genetics and describes the ancestry of the genetic material of individuals in a population experiencing recombination