Occupation densities of Ensembles of Branching Random Walks

We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $lfloor sNrfloor$ branching random walks, viewed as a function-valued, increasing process ${g_{s}^{N}}_{sge 0}$, converges weakly to a pure jump process in the Skorohod space $mathbb D([0, +infty), mathcal C_{0}(mathbb R))$, as $Ntoinfty$. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-valued stable-1/2 subordinator.