# Fragmentations with self-similar branching speeds

## TL;DR

This paper generalizes self-similar fragmentation processes by incorporating marks that evolve as independent positive self-similar Markov processes, providing a Lévy-Khinchin characterization and conditions for finite-time absorption and finite genealogical tree length.

## Contribution

It introduces a new class of fragmentation processes with marked partitions, extending previous models with a Lévy-Khinchin representation and analysis of absorption and genealogical properties.

## Key findings

- Lévy-Khinchin representation for the generalized fragmentation processes
- Conditions for finite-time absorption into a frozen state
- Criteria for finite total length of the genealogical tree

## Abstract

We consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (2002). Our main result is the characterization of these generalized fragmentation processes: a L\'evy-Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.