A Note on the Transience of Critical Branching Random Walks on the Line

Gerold Alsmeyer,Matthias Meiners

doi:10.46298/dmtcs.3581

Abstract

Gantert and Müller (2006) proved that a critical branching random walk (BRW) on the integer lattice is transient by analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic behavior of the left- and rightmost particles in a critical BRW as time goes to infinity is provided in the final section by drawing on recent work by Hu and Shi (2008).

Highlights

Consider a cloud of particles which moves on the line as follows
The daughter particles are independently displaced relative to their mother’s site in accordance with the same step size distribution Q, say. This process continues indefinitely, i.e., each new born particle splits after one unit of time in accordance withj≥0, and the relative displacement of each daughter particle with respect to its mother’s site has distribution Q and is independent of the relative displacements of its siblings as well as of the history of the process. This model describes a special nonextinctive branching random walk (BRW), the specialization being that the relative displacements of siblings are i.i.d. rather than chosen from a general point process on R
In order to describe the positions of all living particles at time n, we introduce the random location measures

Summary

A Note on the Transience of Critical Branching Random Walks on the Line

To cite this version: Gerold Alsmeyer, Matthias Meiners. A Note on the Transience of Critical Branching Random Walks on the Line. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments.

Introduction

Model description and main results

Random weighted location measures and an associated random walk

Extremal particle positions in a critical BRW