Trees within trees II: Nested Fragmentations

TL;DR

This paper extends the theory of fragmentation processes to nested partitions, characterizing their jump measures and identifying new types of erosion and dislocation specific to the nested setting.

Contribution

It generalizes the concept of homogeneous Markov fragmentation processes to nested partitions, providing a detailed characterization of their jump measures.

Findings

Identified three forms of erosion in nested fragmentation.

Discovered two types of dislocation, including a nested-specific one.

Connected nested dislocation to a bivariate paintbox process.

Abstract

Similarly as in (Blancas et al. 2018) where nested coalescent processes are studied, we generalize the definition of partition-valued homogeneous Markov fragmentation processes to the setting of nested partitions, i.e. pairs of partitions $(\zeta,\xi)$ where $\zeta$ is finer than $\xi$. As in the classical univariate setting, under exchangeability and branching assumptions, we characterize the jump measure of nested fragmentation processes, in terms of erosion coefficients and dislocation measures. Among the possible jumps of a nested fragmentation, three forms of erosion and two forms of dislocation are identified - one of which being specific to the nested setting and relating to a bivariate paintbox process.