# Occupation densities of Ensembles of Branching Random Walks

## TL;DR

This paper investigates the asymptotic behavior of occupation densities in large ensembles of critical, driftless branching random walks, revealing convergence to a jump process characterized by integrated super-Brownian excursions.

## Contribution

It establishes the weak convergence of rescaled occupation densities of multiple branching random walks to a pure jump process involving ISE densities and stable subordinators.

## Key findings

- Rescaled occupation densities converge to a pure jump process.
- Jumps are i.i.d. copies of ISE densities.
- Limiting process involves stable-1/2 subordinator.

## 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.