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.