The Brownian web, the Brownian net, and their universality

TL;DR

This paper introduces the Brownian web and net, explaining their roles as scaling limits in various one-dimensional stochastic models with branching and coalescence, and surveys related models and open questions.

Contribution

It provides an accessible introduction to the Brownian web and net, highlighting their universality and connections to diverse models in probability theory.

Findings

Brownian web describes coalescing Brownian motions from all points in space-time.

Brownian net generalizes the web to include branching processes.

Various models like random walks and percolation relate to these structures.

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.