Multifractal Analysis of Refined Sets in Branching Random Walks on Galton–Watson Trees

In this paper, we investigate refined level sets associated with branching random walks on the boundary of a supercritical Galton–Watson tree. More precisely, for a prescribed deterministic sequence s:=(rn,r˜n), we introduce refined deviation sets, denoted by Es(α,β), consisting of boundary points for which the deviations SnX(t)−αSnX˜(t) and SnY(t)−βSnY˜(t) are asymptotically equivalent to rn and r˜n, respectively, as n→∞. This setting extends the classical law-of-large-numbers level sets by allowing for controlled deviations around the typical linear behavior of the branching random walks. By constructing suitable inhomogeneous Mandelbrot measures, we establish sufficient conditions under which the sets Es(α,β) have maximal Hausdorff and packing dimensions.