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The Brownian web.

L R G Fontes1, M Isopi, C M Newman

  • 1Instituto de Matemática e Estatistica, Universidade de São Paulo, 05508-090 São Paulo, Brazil.

Proceedings of the National Academy of Sciences of the United States of America
|November 27, 2002
PubMed
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Researchers constructed the Brownian Web, a novel random process representing coalescing one-dimensional Brownian motions. This framework establishes convergence criteria for random walks, showing their scaling limit approximates the Brownian Web.

Area of Science:

  • Probability theory
  • Stochastic processes
  • Mathematical physics

Background:

  • Previous work by Arratia, Toth, and Werner established random processes for coalescing one-dimensional Brownian motions.
  • These processes formally model systems where multiple random paths merge over time and space.

Purpose of the Study:

  • To extend existing work by constructing and characterizing a new random variable, the Brownian Web.
  • To define the Brownian Web within a suitable metric space of path sets.
  • To establish general convergence criteria for coalescing random walks.

Main Methods:

  • Construction of the Brownian Web as a random variable.
  • Characterization of the Brownian Web in a metric space of compact path sets.
  • Development of convergence criteria for random walks.

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Main Results:

  • The Brownian Web is formally defined and characterized.
  • General convergence criteria are established.
  • Coalescing random walks are shown to converge in distribution to the Brownian Web in the scaling limit.

Conclusions:

  • The Brownian Web provides a unified framework for studying coalescing random processes.
  • The established convergence criteria offer new analytical tools for stochastic systems.
  • This work bridges the gap between discrete random walks and continuous Brownian motion models.