Phase transition of the consistent maximal displacement of branching Brownian motion - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2024

Phase transition of the consistent maximal displacement of branching Brownian motion

Résumé

Consider branching Brownian motion in which we begin with one particle at the origin, particles independently move according to Brownian motion, and particles split into two at rate one. It is well-known that the right-most particle at time $t$ will be near $\sqrt{2} t$. Roberts considered the so-called consistent maximal displacement and showed that with high probability, there will be a particle at time $t$ whose ancestors stayed within a distance $ct^{1/3}$ of the curve $s \mapsto \sqrt{2} s$ for all $s \in [0, t]$, where $c = (3 \pi^2)^{1/3}/\sqrt{2}$. We consider the question of how close the trajectory of a particle can stay to the curve $s \mapsto (\sqrt{2} + \varepsilon) s$ for all $s \in [0, t]$, where $\varepsilon> 0$ is small. We find that there is a phase transition, with the behavior changing when $t$ is of the order $\varepsilon^{-3/2}$. This result allows us to determine, for branching Brownian motion in which particles have a drift to the left of $\sqrt{2} + \varepsilon$ and are killed at the origin, the position at which a particle needs to begin at time zero for there to be a high probability that the process avoids extinction until time $t$.
Fichier principal
Vignette du fichier
Tightv11.pdf (550.17 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04614958 , version 1 (17-06-2024)

Identifiants

Citer

Julien Berestycki, Jiaqi Liu, Bastien Mallein, Jason Schweinsberg. Phase transition of the consistent maximal displacement of branching Brownian motion. 2024. ⟨hal-04614958⟩
0 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More