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The Brownian web, the Brownian net, and their universality

Research paper by Emmanuel Schertzer, Rongfeng Sun, Jan M. Swart

Indexed on: 01 Oct '15Published on: 01 Oct '15Published in: Mathematics - Probability



Abstract

The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They appear in the diffusive scaling limits of many one-dimensional interacting particle systems with branching and coalescence. This article gives an introduction to the Brownian web and net, and how they arise in the scaling limits of various one-dimensional models, focusing mainly on coalescing random walks and random walks in i.i.d. space-time random environments. We will also briefly survey models and results connected to the Brownian web and net, including alternative topologies, population genetic models, true self-repelling motion, planar aggregation, drainage networks, oriented percolation, black noise and critical percolation. Some open questions are discussed at the end.