Total number of births on the negative half-line of the binary branching Brownian motion in the boundary case

Bastien Mallein,Xinxin Chen

doi:10.1214/22-ecp449

Abstract

The binary branching Brownian motion in the boundary case is a particle system on the real line behaving as follows. It starts with a unique particle positioned at the origin at time 0. The particle moves according to a Brownian motion with drift μ=2 and diffusion coefficient σ2=2, until an independent exponential time of parameter 1. At that time, the particle dies giving birth to two children who then start independent copies of the same process from their birth place. It is well-known that in this system, the cloud of particles eventually drifts to ∞. The aim of this note is to provide a precise estimate for the total number of particles that were born on the negative half-line, investigating in particular the tail decay of this random variable.

Highlights

A branching Brownian motion is a continuous-time particle system on the real line in which particles move according to independent Brownian motions and split at independent exponential times into children
We assume the branching Brownian motion to be in the so-called boundary case, i.e. that the Brownian motions driving the motion of the particles have drift μ = 2 and diffusion coefficient σ2 = 2
Jagers [16] proved that branching processes stopped at Lx satisfies the branching property, i.e. that each particle in Lx starts from its time and position an independent copy of the branching Brownian motion, which is independent of σ ((Xs(u), s ≤ t), (u, t) ∈ Lx)

Summary

Introduction

A branching Brownian motion is a continuous-time particle system on the real line in which particles move according to independent Brownian motions and split at independent exponential times into children. Remark that the random variable Nx is related to, but different of, the number N x of births that occurred in the branching Brownian motion with absorption at level x, defined as

Objectives

Results

Conclusion