Nested fragmentations of $\Bbb{N}$

Motivated by the problem of reconstructing species tree from the inference of gene trees in evolutionary biology, I study some exchangeable Markov branching processes, with values in the partitions of $\Bbb{N}$, which can be seen as the joint evolution of a 'tree within a tree'. Under a given set of assumptions, I will give a characterization of their distributions in terms of simpler objects, namely three erosion coefficients and two dislocation measures. We can then construct nested fragmentations as jump processes, where the jumps are according to a time-homogeneous Poisson point process.