Consistent Markov branching trees with discrete edge lengths

Abstract

We study consistent collections of random fragmentation trees with random integer-valued edge lengths. We prove several equivalent necessary and sufficient conditions under which Geometrically distributed edge lengths can be consistently assigned to a Markov branching tree. Among these conditions is a characterization by a unique probability measure, which plays a role similar to the dislocation measure for homogeneous fragmentation processes. We discuss this and other connections to previous work on Markov branching trees and homogeneous fragmentation processes.