The Seneta-Heyde scaling for homogeneous fragmentations

TL;DR

This paper extends the Seneta-Heyde scaling results from branching random walks to homogeneous fragmentation processes, introducing a versatile $L^p$-based approach applicable across various branching-type stochastic models.

Contribution

It provides the first analysis of Seneta-Heyde norming in fragmentation processes and develops a unified $L^p$-based reasoning method for multiple branching models.

Findings

Established Seneta-Heyde norming for homogeneous fragmentations

Developed an $L^p$ estimate-based proof technique

Unified approach applicable to branching random walks, Brownian motion, and Gaussian chaos

Abstract

Homogeneous mass fragmentation processes describe the evolution of a unit mass that breaks down randomly into pieces as time. Mathematically speaking, they can be thought of as continuous-time analogues of branching random walks with non-negative displacements. Following recent developments in the theory of branching random walks, in particular the work of \cite{AShi10}, we consider the problem of the Seneta-Heyde norming of the so-called additive martingale at criticality. Aside from replicating results for branching random walks in the new setting of fragmentation processes, our main goal is to present a style of reasoning, based on $L^p$ estimates, which works for a whole host of different branching-type processes. We show that our methods apply equally to the setting of branching random walks, branching Brownian motion as well as Gaussian multiplicative chaos.