Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Marking (1, 2) points of the Brownian web and applications

C. M. Newman, K. Ravishankar, and E. Schertzer

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Abstract

The Brownian web (BW), which developed from the work of Arratia and then Tóth and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional space–time that arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Two recently introduced extensions of the BW, the Brownian net (BN) constructed by Sun and Swart, and the dynamical Brownian web (DyBW) proposed by Howitt and Warren, are (or should be) scaling limits of corresponding discrete extensions of the DW – the discrete net (DN) and the dynamical discrete web (DyDW). These discrete extensions have a natural geometric structure in which the underlying Bernoulli left or right “arrow” structure of the DW is extended by means of branching (i.e., allowing left and right simultaneously) to construct the DN or by means of switching (i.e., from left to right and vice-versa) to construct the DyDW. In this paper we show that there is a similar structure in the continuum where arrow direction is replaced by the left or right parity of the (1, 2) space–time points of the BW (points with one incoming path from the past and two outgoing paths to the future, only one of which is a continuation of the incoming path). We then provide a complete construction of the DyBW and an alternate construction of the BN to that of Sun and Swart by proving that the switching or branching can be implemented by a Poissonian marking of the (1, 2) points.

Résumé

Le réseau Brownien (BW) construit à partir des travaux de Arratia, de Tòth et de Werner est une collection aléatoire de chemins (avec des points de depart determinés) dans un espace deux-dimensionnel (une dimension en temps et une autre en espace), qui est la limite d’échelle d’un réseau discret (DW) de marches aléatoires coalescentes. Récemment, deux extensions du BW ont été introduites: le filet Brownien (BN), construit par Sun et Swart, et le réseau Brownien dynamique (DyBW), proposé par Howitt et Warren. Ces deux objets sont (ou devraient être) la limite d’échelle de deux extensions naturelles du réseau discret – le filet discret (DN) et le réseau dynamique discret (DyDW). Le DN et le DyDW sont obtenus par une modification de la configuration des “flèches” droites ou gauches qui composent le réseau discret. Pour le DN, un mécanisme de ramification est introduit (en permettant des flèches droites et gauches simultanément) alors que pour le DyDW, la direction des flèches est modifiée (de droite à gauche et vice-versa). Dans cet article, nous montrons qu’il existe une structure géométrique analogue dans le cas continu. Plus précisément, la direction des flèches dans le cas discret est remplacée par la direction des points (1, 2) du réseau Brownien (en un point (1, 2) se trouvent un chemin entrant et deux chemins sortants, l’un d’eux étant la continuation du chemin entrant). Nous montrons que les ramifications et changements de direction peuvent être introduits dans le cas continu par un marquage de type Poisson des points (1, 2). Par l’intermédiaire de ce marquage, nous donnons une construction complète du DyBW et une construction alternative à celle de Sun et Swart du BN.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 2 (2010), 537-574.

Dates
First available in Project Euclid: 11 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1273584134

Digital Object Identifier
doi:10.1214/09-AIHP325

Mathematical Reviews number (MathSciNet)
MR2667709

Zentralblatt MATH identifier
1198.60044

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Keywords
Brownian web Brownian net Dynamical Brownian web Coalescing random walks Poissonian marking Nucleation on boundaries Sticky Brownian motion

Citation

Newman, C. M.; Ravishankar, K.; Schertzer, E. Marking (1, 2) points of the Brownian web and applications. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 2, 537--574. doi:10.1214/09-AIHP325. https://projecteuclid.org/euclid.aihp/1273584134


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