Modulus of continuity of a class of monofractal processes

TL;DR

This paper determines the modulus of continuity for Canonical Embedded Branching Processes (CEBP), establishing their monofractal nature and including Brownian motion as a special case, using adapted techniques from fractal analysis.

Contribution

It introduces the modulus of continuity for CEBP and proves their monofractality, extending analysis methods to non-Markovian processes on fractals.

Findings

CEBP are monofractal processes.

Brownian motion is a special case of CEBP.

Techniques from fractal analysis are adapted for non-Markovian processes.

Abstract

We derive the modulus of continuity of a class of processes called Canonical Embedded Branching Processes (CEBP), recently introduced by Decrouez and Jones, and we establish their monofractal character. CEBP provide a rich class of processes, including the Brownian motion as a particular case. The techniques developed in this study follow the steps of Barlow and Perkins on Brownian motion on a Sierpinski gasket, though complications arise here since CEBP are not Markovian in general.