Stochastic flows in the Brownian web and net

Schertzer, Emmanuel; Sun, Rongfeng; Swart, Jan M.

doi:10.1090/s0065-9266-2013-00687-9

Cited by 26 publications

(47 citation statements)

References 14 publications

(56 reference statements)

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“…Interesting connections with the Brownian web and net have also emerged from many unexpected sources, including supercritical oriented percolation on Z 1+1 [AS11], planar aggregation models [NT12,NT15], true self-avoiding random walks on Z [T95, TW98], random matrix theory [TZ11,TYZ12], and also one-dimensional random walks in i.i.d. space-time random environments [SSS14]. There are also close parallels between the Brownian web and the scaling limit of critical planar percolation, which are the only known examples of two-dimensional black noise [T04a, T04b, SS11, EF16].…”

Section: Introductionmentioning

confidence: 99%

The Brownian Web, the Brownian Net, and their Universality

Schertzer¹,

Sun

Swart

2016

Advances in Disordered Systems, Random Processes and Some Applications

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107

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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.MSC 2000. Primary: 82C21 ; Secondary: 60K35, 60D05.

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Section: Introductionmentioning

confidence: 99%

The Brownian Web, the Brownian Net, and their Universality

Schertzer¹,

Sun

Swart

2016

Advances in Disordered Systems, Random Processes and Some Applications

Self Cite

107

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“…Main distinction of a theorem 1.1 modification is that it combines perfect cocycle property of ϕ with the measurability of the group of shifts θ. It must be noted that a number of various modifications of the Arratia flow that do not deal with the group of shifts of underlying probability space appeared in [8,9,10,11,12].…”

Section: Introductionmentioning

confidence: 99%

Stationary points in coalescing stochastic flows on R

Dorogovtsev¹,

Riabov²,

Schmalfuß³

2020

Stochastic Processes and their Applications

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“…It was proved that the Arratia flow uniquely defines a random compact in the space of compact sets of continuous trajectories. This random compact was called a Brownian Web (we refer to [7,8] for a detailed account on developments related to this very interesting and important object). It was pointed already in [6] that the Brownian Web cannot be viewed as a family of random mappings -with probability 1 there are multiple trajectories in the Brownian Web that start at the same point.…”

Section: Introductionmentioning

confidence: 99%