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

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.