Multifractal spectra and precise rates of decay in homogeneous fragmentation

TL;DR

This paper investigates the multifractal spectra and decay rates in a homogeneous fragmentation process of the unit interval, utilizing additive martingales and Lévy processes to analyze the Hausdorff dimension of specific decay sets.

Contribution

It provides a precise characterization of the Hausdorff dimension for points with exponential decay in a mass-conservative fragmentation, connecting multifractal analysis with Lévy processes.

Findings

Determines the Hausdorff dimension of decay sets.

Links multifractal spectra with Lévy process behavior.

Provides explicit decay rate classifications.

Abstract

We consider a mass-conservative fragmentation of the unit interval. The main purpose of this work is to specify the Hausdorff dimension of the set of locations having exactly an exponential decay. The study relies on an additive martingale which arises naturally in this setting, and a class of L\'evy processes constrained to stay in a finite interval.