Survival of near-critical branching Brownian motion

TL;DR

This paper analyzes the asymptotic behavior of the survival probability in near-critical branching Brownian motion with negative drift, establishing a connection to traveling wave solutions of the Fisher-KPP equation and confirming prior nonrigorous predictions.

Contribution

It provides rigorous asymptotics for the survival probability as the drift approaches criticality, linking probabilistic methods to PDE predictions.

Findings

Existence of a traveling wave limit for survival probability as epsilon approaches zero

Sharp asymptotics of survival probability for large deviations from the critical point

Confirmation of nonrigorous PDE predictions through probabilistic proofs

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 $x \in \R$, $\lim_{\epsilon \to 0} Q_\mu(L+x) = \theta(x) \in (0,1)$ 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.