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Monography Chapter

The Brownian web, the Brownian net, and their universality

Schertzer E., Sun R., Swart Jan M.

: Advances in Disordered Systems, Random Processes and Some Applications, p. 270-368 , Eds: Contucci Pierluigi, Giardina Cristian

: GA15-08819S, GA ČR

: Brownian net, Brownian web, universality

: 10.1017/9781316403877.007


(eng): 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.

: BA

: 10101