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

Charles Newman, K. Ravishankar, E. Schertzer

Research output: Contribution to journalArticle

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 DWis 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.

Original languageEnglish (US)
Pages (from-to)537-574
Number of pages38
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume46
Issue number2
DOIs
StatePublished - May 2010

Fingerprint

Path
Scaling Limit
Sun
Branching
Coalescing Random Walk
Space-time
World Wide Web
Simple Random Walk
Geometric Structure
Bernoulli
Alternate
Parity
Continuation
Continuum
Scaling
Random walk

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Marking (1, 2) points of the Brownian web and applications. / Newman, Charles; Ravishankar, K.; Schertzer, E.

In: Annales de l'institut Henri Poincare (B) Probability and Statistics, Vol. 46, No. 2, 05.2010, p. 537-574.

Research output: Contribution to journalArticle

@article{37e51d7c54fc4314b5619ad2e77b7511,
title = "Marking (1, 2) points of the Brownian web and applications",
abstract = "The Brownian web (BW), which developed from the work of Arratia and then T{\'o}th 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 DWis 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.",
keywords = "Brownian net, Brownian web, Coalescing random walks, Dynamical Brownian web, Nucleation on boundaries, Poissonian marking, Sticky Brownian motion",
author = "Charles Newman and K. Ravishankar and E. Schertzer",
year = "2010",
month = "5",
doi = "10.1214/09-AIHP325",
language = "English (US)",
volume = "46",
pages = "537--574",
journal = "Annales de l'institut Henri Poincare (B) Probability and Statistics",
issn = "0246-0203",
publisher = "Institute of Mathematical Statistics",
number = "2",

}

TY - JOUR

T1 - Marking (1, 2) points of the Brownian web and applications

AU - Newman, Charles

AU - Ravishankar, K.

AU - Schertzer, E.

PY - 2010/5

Y1 - 2010/5

N2 - 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 DWis 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.

AB - 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 DWis 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.

KW - Brownian net

KW - Brownian web

KW - Coalescing random walks

KW - Dynamical Brownian web

KW - Nucleation on boundaries

KW - Poissonian marking

KW - Sticky Brownian motion

UR - http://www.scopus.com/inward/record.url?scp=77952574312&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952574312&partnerID=8YFLogxK

U2 - 10.1214/09-AIHP325

DO - 10.1214/09-AIHP325

M3 - Article

AN - SCOPUS:77952574312

VL - 46

SP - 537

EP - 574

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 2

ER -