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Stochastic Flows in the Brownian Web and Net - Memoirs of the American Mathematical Society (Paperback)
  • Stochastic Flows in the Brownian Web and Net - Memoirs of the American Mathematical Society (Paperback)

Stochastic Flows in the Brownian Web and Net - Memoirs of the American Mathematical Society (Paperback)

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Paperback 160 Pages / Published: 30/01/2014
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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.

Publisher: American Mathematical Society
ISBN: 9780821890882
Number of pages: 160
Dimensions: 254 x 178 mm

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