Multifractal analysis and Erd\"os-R\'enyi laws of large numbers for branching random walks in $\R^d$

TL;DR

This paper conducts a detailed multifractal analysis of $ ^d$-valued branching random walks, establishing Erd"os-Renyi laws of large numbers on various level sets, including critical and boundary cases, under finite exponential moment conditions.

Contribution

It extends multifractal analysis to boundary and critical levels of branching random walks, incorporating the influence of the boundary metric and providing a comprehensive framework.

Findings

Established Erd"os-Renyi LLN for interior levels

Extended control to boundary levels with strengthened assumptions

Decomposed boundary into convex sets for analysis

Abstract

We revisit the multifractal analysis of $\R^d$-valued branching random walks averages by considering subsets of full Hausdorff dimension of the standard level sets, over each infinite branch of which a quantified version of the Erd\"os-R\'enyi law of large numbers holds. Assuming that the exponential moments of the increments of the walks are finite, we can indeed control simultaneously such sets when the levels belong to the interior of the compact convex domain $I$ of possible levels, i.e. when they are associated to so-called Gibbs measures, as well as when they belong to the subset $(\partial{I})_{\mathrm{crit}}$ of $\partial I$ made of levels associated to ``critical'' versions of these Gibbs measures. It turns out that given such a level of one of these two types, the associated Erd\"os-R\'enyi LLN depends on the metric with which is endowed the boundary of the underlying Galton-Watson tree. To extend our control to all the boundary points in cases where $\partial I\neq (\partial{I})_{\mathrm{crit}}$, we slightly strengthen our assumption on the distribution of the increments to exhibit a natural decomposition of $\partial I\setminus (\partial{I})_{\mathrm{crit}}$ into at most countably many convex sets $J$ of affine dimension $\le d-1$ over each of which we can essentially reduce the study to that of interior and critical points associated to some $\R^{\dim J}$-valued branching random~walk.