Yaglom theorem for critical branching random walk on $\mathbb{Z}^d$

TL;DR

This paper investigates the asymptotic behavior of occupation times for critical branching random walks on integer lattices, revealing dimension-dependent scaling laws and establishing weak convergence results.

Contribution

It provides the first detailed analysis of occupation time asymptotics for critical branching random walks conditioned on hitting a set, answering a question posed by Le Gall and Merle.

Findings

Occupation time scales as rac{rac{ ext{ in dimensions }d extless=3

Order of occupation time is rac{ ext{ in dimension }d=4

Occupation time remains bounded in dimensions }d extgreater=5

Abstract

We study the critical branching random walk on $\mathbb{Z}^d$ started from a distant point $x$ and conditioned to hit some compact set $K$ in $\mathbb{Z}^d$. We are interested in the occupation time in $K$ and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order $\|x\|^{4-d}$ in dimensions $d\leq 3$, of order $\log\|x\|$ in dimension $d=4$, and of order 1 in dimensions $d\geq 5$. The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle (Elect. Comm. in Probab. 11 (2006), 252-265).