Occupation densities of ensembles of branching random walks

Si Tang,Steven P Lalley

doi:10.1214/20-ecp293

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.

Highlights

In a branching random walk on the integers, individuals live for one generation, reproduce as in a Galton-Watson process, giving rise to offspring which independently jump according to the law of a random walk
A branching random walk is said to be critical if the offspring distribution ν has mean 1, and driftless if the jump distribution F has mean 0 and finite variance
We will assume throughout that (i) the offspring distribution ν has mean one and finite, positive variance σν2; and (ii) the step distribution F for the random walk has span one, mean zero and finite, positive variance σF2

Summary

Introduction

In a branching random walk on the integers, individuals live for one generation, reproduce as in a Galton-Watson process, giving rise to offspring which independently jump according to the law of a random walk. Given the labeled tree T associated with the branching random walk, the occupation measures can be recovered as follows. Y s (t, x) dt where {Y s(t, x), x ∈ R}t≥0 is the density process for a super-Brownian motion {Yts}t≥0 with variance parameters (σν2, σF2 ), started from the initial measure Y0s = sδ0.

Results

Conclusion