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

TL;DR

This paper investigates the distribution and tail decay of the total number of particles born on the negative half-line in a boundary case binary branching Brownian motion, providing precise estimates for this quantity.

Contribution

It offers a detailed analysis and precise estimates of the tail behavior of the total number of particles born on the negative half-line in this specific branching Brownian motion model.

Findings

The total number of particles on the negative half-line has a heavy tail decay.

Explicit asymptotic estimates for the tail distribution are derived.

The results enhance understanding of boundary behaviors in branching processes.

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 $\mu = 2$ and diffusion coefficient $\sigma^2 = 2$, 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 $\infty$. 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.