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

Xinxin Chen (1, 2) , Bastien Mallein (3, 4)

1 ICJ - Institut Camille Jordan
2 PSPM - Probabilités, statistique, physique mathématique
3 LAGA - Laboratoire Analyse, Géométrie et Applications
4 DMA - Département de Mathématiques et Applications - ENS Paris

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.

Domaines: Probabilités [math.PR]

Xinxin Chen, Bastien Mallein. Total number of births on the negative half-line of the binary branching Brownian motion in the boundary case. Electronic Communications in Probability, 2022, 27 (7), pp.1-11. ⟨10.1214/22-ECP449⟩. ⟨hal-03251329v2⟩