Branching Brownian Motion with Catalytic Branching at the Origin (original) (raw)

On the spread of a branching Brownian motion whose offspring number has infinite variance

We study the impact on shape parameters of an underlying Bienaym\'e-Galton-Watson branching process (height, width and first hitting time), of having a non-spatial branching mechanism with infinite variance. Aiming at providing a comparative study of the spread of an epidemics whose dynamics is given by the modulus of a branching Brownian motion (BBM) we then consider spatial branching processes in dimension d, not necessarily integer. The underlying branching mechanism is then either a binary branching model or one presenting infinite variance. In particular we evaluate the chance p(x) of being hit if the epidemics started away at distance x. We compute the large x tail probabilities of this event, both when the branching mechanism is regular and when it exhibits very large fluctuations. Online first, 9 Feb 2015, Physica D Nonlinear Phenomena, under the new title: On extreme events for non-spatial and spatial branching Brownian motions.

The unscaled paths of branching Brownian motion

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2012

For a set A ⊂ C[0, ∞), we give new results on the growth of the number of particles in a branching Brownian motion whose paths fall within A. We show that it is possible to work without rescaling the paths. We give large deviations probabilities as well as a more sophisticated proof of a result on growth in the number of particles along certain sets of paths. Our results reveal that the number of particles can oscillate dramatically. We also obtain new results on the number of particles near the frontier of the model. The methods used are entirely probabilistic. Résumé. Considérons un mouvement Brownien branchant. Nous nous intéressons au nombre de particules dont le chemin reste dans un ensemble fixé A ⊂ C[0, ∞). Nous montrons qu'il n'est pas nécessaire de renormaliser les chemins. Nous donnons les probabilités de grandes déviations, ainsi qu'une preuve plus sophistiquée pour un résultat concernant la croissance du nombre de particules dans certains ensembles. Nos résultats démontrent que ce nombre de particules peut fortement osciller. Nous obtenons aussi des résultats nouveaux concernant le nombre de particules proches de la frontière du système. Nos méthodes sont purement probabilistes.

Anomalous spreading in reducible multitype branching Brownian motion

Electronic Journal of Probability, 2021

We consider a two-type reducible branching Brownian motion, defined as a two type branching particle system on the real line, in which particles of type 1 can give birth to particles of type 2, but not reciprocally. This process has been shown by Biggins [Big12] to exhibit an anomalous spreading behaviour under specific conditions: in that situation, the rightmost particle at type t is much further than the expected position for the rightmost particle in a branching Brownian motion consisting only of particles of type 1 or of type 2. This anomalous spreading also has been investigated from a reaction-diffusion equation standpoint by Holzer [Hol14,. The aim of this article is to refine the previous results and study the asymptotic behaviour of the extremal process of the two-type reducible branching Brownian motion. If the branching Brownian motion exhibits an anomalous spreading behaviour, its asymptotic differs from what it typically expected in branching Brownian motions.

Poissonian statistics in the extremal process of branching Brownian motion

The Annals of Applied Probability, 2012

As a first step towards a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [1] that, in the limit of large time t, extremal particles descend with overwhelming probability from ancestors having split either within a distance of order one from time 0, or within a distance of order one from time t. The result suggests that the extremal process of branching Brownian motion is a randomly shifted cluster point process. Here we put part of this picture on rigorous ground: we prove that the point process obtained by retaining only those extremal particles which are also maximal inside the clusters converges in the limit of large t to a random shift of a Poisson point process with exponential density. The last section discusses the Tidal Wave Conjecture by Lalley and Sellke [18] on the full limiting extremal process and its relation to the work of Chauvin and Rouault [9] on branching Brownian motion with atypical displacement.

Genealogy of extremal particles of branching Brownian motion

Communications on Pure and Applied Mathematics, 2011

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 Fisher-KPP equation to traveling waves , the law of the rightmost particle in the limit of large times is rather well understood. In this work, we address the full statistics of the extremal particles (first-, second-, third-etc. largest). In particular, we prove that in the large t−limit, such particles descend with overwhelming probability from ancestors having split either within a distance of order one from time 0, or within a distance of order one from time t. The approach relies on characterizing, up to a certain level of precision, the paths of the extremal particles. As a byproduct, a heuristic picture of Branching Brownian Motion "at the edge" emerges, which sheds light on the still unknown limiting extremal process.

A conceptual approach to a path result for branching Brownian motion

Stochastic Processes and their Applications, 2006

This article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author's benefit and for the benefit of the author's institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues that you know, and providing a copy to your institution's administrator.

Survival probabilities for branching Brownian motion with absorption

Electronic Communications in Probability, 2007

We study a branching Brownian motion (BBM) with absorption, in which particles move as Brownian motions with drift -ρ, undergo dyadic branching at rate β > 0, and are killed on hitting the origin. In the case ρ > √ 2β the extinction time for this process, ζ, is known to be finite almost surely. The main result of this article is a large-time asymptotic formula for the survival probability P x (ζ > t) in the case ρ > √ 2β, where P x is the law of the BBM with absorption started from a single particle at the position x > 0. We also introduce an additive martingale, V , for the BBM with absorption, and then ascertain the convergence properties of V . Finally, we use V in a 'spine' change of measure and interpret this in terms of 'conditioning the BBM to survive forever' when ρ > √ 2β, in the sense that it is the large t-limit of the conditional probabilities P x (A|ζ > t + s), for A ∈ F s .

The extremal process of a cascading family of branching Brownian motion

arXiv (Cornell University), 2022

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 move on the real line according to Brownian motions and branch at rate 1 into two children of type 1. Furthermore, at rate α, they give birth to children too of type 2. Particles of type 2 move according to standard Brownian motion and branch at rate 1, but cannot give birth to descendants of type 1. We obtain the asymptotic behaviour of the extremal process of particles of type 2.

Branching Brownian motion in a strip: Survival near criticality

The Annals of Probability, 2016

We consider a branching Brownian motion with linear drift in which particles are killed on exiting the interval (0, K) and study the evolution of the process on the event of survival as the width of the interval shrinks to the critical value at which survival is no longer possible. We combine spine techniques and a backbone decomposition to obtain exact asymptotics for the near-critical survival probability. This allows us to deduce the existence of a quasi-stationary limit result for the process conditioned on survival which reveals that the backbone thins down to a spine as we approach criticality. This paper is motivated by recent work on survival of near critical branching Brownian motion with absorption at the origin by Aïdékon and Harris [Near-critical survival probability of branching Brownian motion with an absorbing barrier (2010) Unpublished manuscript] as well as the work of Berestycki et al.

An ergodic theorem for the frontier of branching Brownian motion

Electronic Journal of Probability, 2013

We prove a conjecture of Lalley and Sellke [Ann. Probab. 15 (1987)] asserting that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a double exponential, or Gumbel, distribution with a random shift. The method of proof is based on the decorrelation of the maximal displacements for appropriate time scales. A crucial input is the localization of the paths of particles close to the maximum that was previously established by the authors [Comm. Pure Appl. Math. 64 ].