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

Najmeddine Attia ()
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Najmeddine Attia: Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Mathematics, 2025, vol. 13, issue 17, 1-24

Abstract: In the present work, we consider three branching random walk S n Z ( t ) , Z ∈ { X , Y , Φ } on a supercritical random Galton–Watson tree ∂ T . We compute the Hausdorff and packing dimensions of the level set E χ ( α , β ) = t ∈ ∂ T : lim n → ∞ S n X ( t ) S n Y ( t ) = α and lim n → ∞ S n Y ( t ) n = β , where ∂ T is endowed with random metric using S n Φ ( 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.

Keywords: branching random walk; relative multifractal formalism; multifractal measures; Hausdorff and packing dimensions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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