Occupation densities of Ensembles of Branching Random Walks

Abstract

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 $\lfloor sN\rfloor$ branching random walks, viewed as a function-valued, increasing process $\{g_s^N{\}}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb{D}([0,+\infty ),{\mathcal{C}}_0(\mathbb{R}))$, as $N\to \infty$. 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.

12 pages, 1 figure