  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.
