The Brownian Net

In probability theory, the Brownian net is a continuum object that describes the scaling limit of coalescing and branching random walks. It extends the better known Brownian web that does not have branching.

The Brownian web was introduced in the year 2004 by the mathematicians Fontes, Isopi, Newman a Ravishankar, who built on earlier work of Arratia, Tóth, and Werner. Informally, it can be described as an uncountable collection of coalescing Brownian motions starting from each point in space-time. Since its introduction it has been studied intensively.

A system of coalescing Brownian motions with branching. Time is plotted vertically.A system of coalescing Brownian motions with branching. Time is plotted vertically.

The Brownian net is an extension of the Brownian web that includes branching. Naively, we cam imagine it as a collection of particles that move in space as Brownian motions. Particles coalesce as soon as they meet and in addition multiply. The majority of newly produced particles immediately coalesces with the particle they originate from, but because the branching rate is high - in fact infinite in the limit - some particles manage to escape their parents.

A system of coalescing random walks with branching. The blue and red paths converge to Brownian motions with drift to the left and right, respectively.A system of coalescing random walks with branching. The blue and red paths converge to Brownian motions with drift to the left and right, respectively.

The Brownian web and net are mainly of theoretical importance in spite of attempts by some authors to use them in the description of river basins. A more theoretical application is their use in the description of stochastic flows. From a mathematical point of view they are complicated objects and many problems relating to them remain unresolved.

An example of a Howitt-Warren flow, which is a stochastic flow whose properties have been investigated with the help of the Brownian web and net. Time is plotted vertically. The picture shows a measure-valued Markov process. The greyscale indicates the intensity of the measure.An example of a Howitt-Warren flow, which is a stochastic flow whose properties have been investigated with the help of the Brownian web and net. Time is plotted vertically. The picture shows a measure-valued Markov process. The greyscale indicates the intensity of the measure.

Related publications:

  1. SUN, Rongfeng; SWART, Jan M. The brownian net. 2008.
  2. SCHERTZER, Emmanuel; SUN, Rongfeng; SWART, Jan. Stochastic flows in the Brownian web and net. American mathematical society, 2014.
  3. SCHERTZER, Emmanuel; SUN, Rongfeng; SWART, Jan M. The Brownian web, the Brownian net, and their universality. Advances in disordered systems, random processes and some applications, 2017, 270-368.
     

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Jan Swart

 

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