Branching Random Walks on Multi-Type Galton–Watson Trees: A Multifractal Analysis

We study the multifractal structure of a branching random walk on the boundary of a finite-type Galton–Watson tree. For α∈R, we consider the level set EX(α)=t∈∂T:limn→∞SnX(t)n=α. The multi-type structure leads naturally to the weighted mean offspring matrix M(q) and to the deterministic pressure P˜(q)=logρ(M(q)). Using the corresponding Perron–Frobenius eigenvectors, we construct a family of multi-type Mandelbrot martingales and prove their almost sure and Lp uniform convergence on compact subsets of the admissible parameter domain. This yields the associated Mandelbrot measures and the uniform estimates required for the multifractal analysis. As a consequence, for every α in a suitable admissible set I, we obtain almost surely dimHEX(α)=P˜*(α), where P˜* denotes the Legendre transform of P˜ and dimH stands for the Hausdorff dimension. Thus the classical multifractal formalism for one-type branching random walks extends to the genuinely multi-type setting, with the scalar normalization replaced by the Perron–Frobenius spectral structure of the weighted offspring matrices.