Self-similar signed growth-fragmentations

TL;DR

This paper extends growth-fragmentation models to include signed masses, introduces genealogical martingales and spinal decomposition, and studies a specific self-similar signed process related to cutting half-planar excursions.

Contribution

It introduces the theory of signed growth-fragmentation processes, including genealogical martingales and spinal decomposition, and analyzes a specific self-similar signed process from half-planar excursions.

Findings

Extended growth-fragmentation theory to signed masses.

Established genealogical martingales and spinal decomposition.

Connected the process to previously studied positive mass models.

Abstract

Growth-fragmentation processes model the evolution of positive masses which undergo binary divisions. The aim of this paper is twofold. First, we extend the theory of growth-fragmentation processes to allow signed mass. Among others, we introduce genealogical martingales and establish a spinal decomposition for the associated cell system, following arXiv:1605.00581. Then, we study a particular family of such self-similar signed growth-fragmentation processes which arise when cutting half-planar excursions at horizontal levels. When restricting this process to the positive masses, we recover part of the family introduced by Bertoin, Budd, Curien and Kortchemski in arXiv:1605.00581.