Survival of near-critical branching Brownian motion

Abstract

Consider a system of particles performing branching Brownian motion with negative drift $\mu =\sqrt{2-\epsilon }$ and killed upon hitting zero. Initially there is one particle at $x>0$. Kesten showed that the process survives with positive probability if and only if $\epsilon >0$. Here we are interested in the asymptotics as $\eps\to 0$ of the survival probability $Q_{\mu }(x)$. It is proved that if $L=\pi /\sqrt{\epsilon }$ then for all $x\in \mathbb{R}$, ${\lim }_{\epsilon \to 0}{Q}_{\mu }(L+x)=\theta (x)\in (0,1)$ exists and is a travelling wave solution of the Fisher-KPP equation. Furthermore, we obtain sharp asymptotics of the survival probability when $x<L$ and $L-x\to \infty$. The proofs rely on probabilistic methods developed by the authors in a previous work. This completes earlier work by Harris, Harris and Kyprianou and confirms predictions made by Derrida and Simon, which were obtained using nonrigorous PDE methods.

References in corpus (2)

• The survival probability of a branching random walk in presence of an absorbing wall
• Quasi-stationary regime of a branching random walk in presence of an absorbing wall

Cited by in corpus (5)

• The genealogy of branching Brownian motion with absorption
• Speed and fluctuations of N-particle branching Brownian motion with spatial selection
• An invariance principle for branching diffusions in bounded domains
• Last passage percolation and traveling fronts
• A Yaglom type asymptotic result for subcritical branching Brownian motion with absorption