# The extremal process of two-speed branching Brownian motion

@article{Bovier2013TheEP,
title={The extremal process of two-speed branching Brownian motion},
author={Anton Bovier and Lisa Hartung},
journal={Electronic Journal of Probability},
year={2013},
volume={19},
pages={1-28}
}
• Published 8 August 2013
• Mathematics
• Electronic Journal of Probability
We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is $\sigma_1$ for $s\leq bt$ and $\sigma_2$ when $bt\leq s\leq t$. In the case $\sigma_1>\sigma_2$, the process is the concatenation of two BBM extremal processes, as expected. In the case $\sigma_1<\sigma_2$, a new family  of cluster point processes arises…
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## References

SHOWING 1-10 OF 24 REFERENCES
Poissonian statistics in the extremal process of branching Brownian motion
• Mathematics
• 2012
As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of
The extremal process of branching Brownian motion
• Mathematics
• 2011
AbstractWe prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson
Maximal displacement of branching brownian motion
It is shown that the position of any fixed percentile of the maximal displacement of standard branching Brownian motion in one dimension is 21/2t–3 · 2−3/2 log t + O(1) at time t, the second-order
Supercritical Branching Brownian Motion and K-P-P Equation In the Critical Speed-Area
• Mathematics
• 1990
If Rt is the position of the rightmost particle at time t in a one dimensional branching brownian motion, whore α is the inverse of the mean life time and m is the mean of the
Branching Brownian motion seen from its tip
• Mathematics
• 2011
It has been conjectured since the work of Lalley and Sellke (Ann. Probab., 15, 1052–1061, 1987) that branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an
Slowdown for Time Inhomogeneous Branching Brownian Motion
• Mathematics
• 2012
We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from
Branching brownian motion seen from its left-most particule
This paper is a review of the recent progresses on the branching brownian motion seen from its left-most particule. We describe in particular the recent works by Arguin-Bovier-Kistler and
Branching random walks in time inhomogeneous environments
• Mathematics
• 2011
We study the maximal displacement of branching random walks in a class of time inhomogeneous environments. Specifically, binary branching random walks with Gaussian increments will be considered,
A Conditional Limit Theorem for the Frontier of a Branching Brownian Motion
• Mathematics
• 1987
On demontre un theoreme limite faible qui relie le comportement, pour une grande duree du maximum d'un mouvement brownien ramifie a la valeur limite d'une certaine martingale associee