Conditioning super-Brownian motion on its boundary statistics, and fragmentation (original) (raw)

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.

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.

Some properties of the exit measure for super Brownian motion

Probability Theory and Related Fields, 2002

We consider the exit measure of super-Brownian motion with a stable branching mechanism of a smooth domain D of R d . We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension. This completes the results given by Sheu 20] and generalizes the results of Abraham and Le Gall 2]. We give also the Hausdor dimension of the exit measure and show it is totally disconnected in high dimension. Eventually we prove the exit measure is singular with respect to the surface measure on @D in the critical dimension. Our main tool is the subordinated Brownian snake introduced

A conceptual approach to a path result for branching Brownian motion

Stochastic Processes and their Applications, 2006

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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.

Rescaled Particle Systems Converging to Super-Brownian Motion

Perplexing Problems in Probability, 1999

Super-Brownian motion was originally constructed as a scaling limit of branching random walk. Here we describe recent results which show that, in two or more dimensions, it is also the limit of long range contact processes and long, short, and medium range voter models. Super-Brownian motion and its close relatives have arisen in a variety of different contexts in the last few years. In this article we focus on their appearance as limits of rescaled interacting particle systems and, in particular, on the results in Mueller and Tribe (1995), Durrett and Perkins (1998), and Cox, Durrett and Perkins (1998). Rather than providing complete proofs, which at present are still rather lengthy in some cases, we will focus on explaining why these theorems hold. These results are related in spirit, if not methodology, to recent work of Derbez and Slade (1998) on the convergence of "sufficiently spread out" rescaled lattice trees to Integrated Super-Excursion (ISE) in more than 8 dimensions, and to ongoing work of Derbez, van der Hofstad and Slade on convergence of sufficiently spread out oriented percolation to super-Brownian motion in more than 4 spatial dimensions. These results are described by Slade (1999) elsewhere in this volume. Super-Brownian motion arises as the limit of rescaled branching random walk and so is "trivial" in the mathematical physics terminology. In this same parlance one expects rescaled interacting systems to converge to this trivial limit above a critical dimension and at the critical dimension, perhaps with logarithmic corrections. For the particle systems we will consider in detail, this dimension is two, the critical dimension for recurrence of simple random walk or Brownian motion. We note that for the contact process this critical dimension (as opposed to d = 4 as in the above work on critical oriented percolation) arises because of our long range scaling in this setting. In order to describe super-Brownian motion we start with

A characterization ofh-Brownian motion by its exit distributions

Probability Theory and Related Fields, 1992

Let X h be an h-Brownian motion in the unit ball D c R d with h harmonic, such that the representing measure of h is not singular with respect to the surface measure on 0D. If Y is a continuous strong Markov process in D with the same killing distributions as X h, then Yis a time change of X h. Similar results hold in simply connected domains in C provided with either the Martin or the Euclidean boundary.

Branching Processes - A General Concept

Latin American Journal of Probability and Mathematical Statistics, 2021

The paper has four goals. First, we want to generalize the classical concept of the branching property so that it becomes applicable for historical and genealogical processes (using the coding of genealogies by ($V$-marked) ultrametric measure spaces leading to state spaces mathbbU\mathbb{U}mathbbU resp. mathbbUV\mathbb{U}^VmathbbUV). The processes are defined by well-posed martingale problems. In particular we want to complement the corresponding concept of infinite divisibility developed in \cite{infdiv} for this context. Second one of the two main points, we want to find a corresponding characterization of the generators of branching processes more precisely their martingale problems which is both easy to apply and general enough to cover a wide range of state spaces. As a third goal we want to obtain the branching property of the mathbbU\mathbb{U}mathbbU-valued Feller diffusion respectively mathbbUV\mathbb{U}^VmathbbUV-valued super random walk and the historical process on countable geographic spaces the latter as two examples of...

The extremal process of branching Brownian motion

Probability Theory and Related Fields, 2013

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 clusters is a Poisson process with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform "unusually large displacements", and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov-Petrovsky-Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process, which fully captures the large time asymptotics of the extremal process. The analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses.