On the tail of the branching random walk local time

Antal Járai,Omer Angel,Tom Hutchcroft

doi:10.1007/s00440-020-01014-4

Abstract

Consider a critical branching random walk on mathbb Z^d, dge 1, started with a single particle at the origin, and let L(x) be the total number of particles that ever visit a vertex x. We study the tail of L(x) under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.

Highlights

In this paper we study the tail of the number of times a critical branching random walk on Zd returns to the origin
The result is most interesting in the upper-critical dimension d = 4, where we find that the local time has a stretched-exponential tail
Each particle splits into a random number of offspring particles independently at random according to the offspring distribution μ, and each offspring particle immediately performs an independent simple random walk step

Summary

Introduction

In this paper we study the tail of the number of times a critical branching random walk on Zd returns to the origin. Theorem 1.2 Let d ≥ 1, let (Bn)n≥0 be a branching random walk on Zd whose offspring distribution is critical, non-trivial, and sub-exponential, started with a single particle at the origin, and let L(x) be the total number of particles that visit x. Remark 1.4 In the context of super-Brownian motion (which is a continuum analogue of critical branching random walk), Le Gall and Merle [15] studied the conditional distribution of the occupation measure Z(B1(x)) of the unit ball B1(x) for large x, given that this measure is positive. Their results are closely related to Theorem 1.2. We restrict attention to the usual nearest-neighbour random walk on Zd for clarity of exposition

Branching random walk

Random walk estimates

Diagrammatic expansion of moments

Low dimensions

High dimensions

The critical dimension