Related papers: Extremal process for irreducible multitype branchi…
We 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 cluster process, where the positions of the…
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…
We study a class of multitype branching L\'evy processes, where particles move according to type-dependent L\'evy processes, switch types via an irreducible Markov chain, and branch according to type-dependent laws. This framework…
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 large time $t$, extremal particles…
We prove the convergence of the extremal processes for variable speed branching Brownian motions where the "speed functions", that describe the time-inhomogeneous variance, lie strictly below their concave hull and satisfy a certain weak…
We consider the limiting extremal process ${\mathcal X}$ of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale $Z$, the rescaled "density" of particles, which are…
We study the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. Then we prove the existence of pulsating travelling wave solutions of the corresponding F-KPP equation in…
In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal…
We study the asymptotic behaviour of the extremal process of a cascading family of branching Brownian motions. This is a particle system on the real line such that each particle has a type in addition to his position. Particles of type $1$…
In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is…
We study the long-time behavior of a triangular system of Fisher--KPP type with $k$ interacting components, associated with a reducible multitype branching Brownian motion with $k$ types of particles. For this cascading system, we prove…
In this paper, we study branching Brownian motion with absorption, in which particles undergo Brownian motions with drift and are killed upon reaching the origin. We prove that the extremal process of this branching Brownian motion with…
Using one-dimensional branching Brownian motion in a periodic environment, we give probabilistic proofs of the asymptotics and uniqueness of pulsating travelling waves of the F-KPP equation in a periodic environment. This paper is a sequel…
We extend the results of Arguin et al and A\"\i{}d\'ekon et al on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson…
We consider a two-speed branching random walk, which consists of two macroscopic stages with different reproduction laws. We prove that the centered maximum converges in law to a Gumbel variable with a random shift and the extremal process…
We study the limiting extremal and cluster point processes of branching Brownian motion. The former records the heights of all extreme values of the process, while the latter records the relative heights of extreme values in a genealogical…
Branching Brownian Motion describes a system of particles which diffuse in space and split into offsprings according to a certain random mechanism. In virtue of the groundbreaking work by M. Bramson on the convergence of solutions of the…
It has been conjectured since the work of Lalley and Sellke (1987) that the branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be…
Flip-flop processes refer to a family of stochastic fluid processes which converge to either a standard Brownian motion (SBM) or to a Markov modulated Brownian motion (MMBM). In recent years, it has been shown that complex distributional…
We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via change of measure that penalises particles spending time in an $\e$-neighbourhood of each other. We derive a simplified version of the…