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September 16, 2025Open Access

New Insights into the Multifractal Formalism of Branching Random Walks on Galton–Watson Tree

Key Points

The study computes the Hausdorff and packing dimensions for branching random walks on a random Galton-Watson tree.
Using a Mandelbrot measure, the formalism leads to significant insights into the structure of level sets in random metrics.
Formalizing the approaches from Olsen and Peyrière adds to the understanding of measures and vectors in this context.
This work highlights the complexity of supercritical branching processes, with implications for stochastic processes.

Abstract

In the present work, we consider three branching random walk SnZ (t), Z∈X, Y, Φ on a supercritical random Galton–Watson tree ∂T. We compute the Hausdorff and packing dimensions of the level set Eχ (α, β) =t∈∂T: limn→∞SnX (t) SnY (t) =αandlimn→∞SnY (t) n=β, where ∂T is endowed with random metric using SnΦ (t). This is achieved by constructing a suitable Mandelbrot measure supported on E (α, β). In the case where Φ=1, we develop a formalism that parallels Olsen’s framework (for measures) and Peyrière’s framework (for the vectorial case) within our setting.

New Insights into the Multifractal Formalism of Branching Random Walks on Galton–Watson Tree

Key Points

Abstract

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