Stochastic flows in the Brownian web and net
Author(s)
Bibliographic Information
Stochastic flows in the Brownian web and net
（Memoirs of the American Mathematical Society, no. 1065）
American Mathematical Society, c2013
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"Volume 227, number 1065 (first of 4 numbers), January 2014"
References: p. 153-155
Includes index
Description and Table of Contents
Description
It is known that certain one-dimensional nearest-neighbour random walks in i.i.d. random space-time environments have diffusive scaling limits. Here, in the continuum limit, the random environment is represented by a ‘stochastic flow of kernels', which is a collection of random kernels that can be loosely interpreted as the transition probabilities of a Markov process in a random environment. The theory of stochastic flows of kernels was first developed by Le Jan and Raimond, who showed that each such flow is characterised by its $n$-point motions. The authors' work focuses on a class of stochastic flows of kernels with Brownian $n$-point motions which, after their inventors, will be called Howitt-Warren flows.
The authors' main result gives a graphical construction of general Howitt-Warren flows, where the underlying random environment takes on the form of a suitably marked Brownian web. This extends earlier work of Howitt and Warren who showed that a special case, the so-called “erosion flow'', can be constructed from two coupled “sticky Brownian webs''. The authors' construction for general Howitt-Warren flows is based on a Poisson marking procedure developed by Newman, Ravishankar and Schertzer for the Brownian web. Alternatively, the authors show that a special subclass of the Howitt-Warren flows can be constructed as random flows of mass in a Brownian net, introduced by Sun and Swart. Using these constructions, the authors prove some new results for the Howitt-Warren flows.
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