%0 Generic %1 berestycki2010nearcritical %A Berestycki, Julien %A Berestycki, Nathanaël %A Schweinsberg, Jason %D 2010 %K Fisher-KPP branching_Brownian_motion travelling_wave %T Survival of near-critical branching Brownian motion %U http://arxiv.org/abs/1009.0406 %X Consider a system of particles performing branching Brownian motion with negative drift $= 2 - \epsilon$ 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 $\eps0$ of the survival probability $Q_\mu(x)$. It is proved that if $L= \pi/\epsilon$ then for all $x ın \R$, $łim_0 Q_\mu(L+x) = þeta(x) (0,1)$ 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 $L-x ınfty$. 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.

@misc{berestycki2010nearcritical, 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. }, added-at = {2010-09-03T17:56:49.000+0200}, author = {Berestycki, Julien and Berestycki, Nathana\"el and Schweinsberg, Jason}, biburl = {https://www.bibsonomy.org/bibtex/24701410e82084ea59b8b1e5954e5642d/peter.ralph}, description = {[1009.0406] Survival of near-critical branching Brownian motion}, interhash = {219ac01acc9c8d75276644932c20dd25}, intrahash = {4701410e82084ea59b8b1e5954e5642d}, keywords = {Fisher-KPP branching_Brownian_motion travelling_wave}, note = {cite arxiv:1009.0406 }, timestamp = {2013-09-12T22:23:01.000+0200}, title = {Survival of near-critical branching Brownian motion}, url = {http://arxiv.org/abs/1009.0406}, year = 2010 }