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Stochastic flows in the Brownian web and net / Emmanuel Schertzer, Rongfeng Sun and Jan M. Swart.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical Society ; v 227, no 1065.Publication details: Providence : American Mathematical Society, c2014.Description: vi, 160 p. : illustrations ; 26 cmISBN:
  • 9780821890882 (pbk. : alk. paper)
Subject(s): DDC classification:
  • 510 23 Am512
Contents:
1. Introduction -- 2. Results for Howitt-Warren flows -- 3. Construction of Howitt-Warren flows in the Brownian web -- 4. Construction of Howitt-Warren flows in the Brownian net -- 5. Outline of the proofs -- 6. Coupling of the Bownian web and net -- 7. Construction and convergence of Howitt-Warren flows -- 8. Support properties -- 9. Atomic or non-atomic -- 10. Infinite starting mass and discrete approximation -- 11. Ergodic properties-- Appendices-- References-- Index.
Summary: 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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"Volume 227, number 1065 (first of 4 numbers), January 2014."

Includes bibliographical references (pages 153-158) and index.

1. Introduction --
2. Results for Howitt-Warren flows --
3. Construction of Howitt-Warren flows in the Brownian web --
4. Construction of Howitt-Warren flows in the Brownian net --
5. Outline of the proofs --
6. Coupling of the Bownian web and net --
7. Construction and convergence of Howitt-Warren flows --
8. Support properties --
9. Atomic or non-atomic --
10. Infinite starting mass and discrete approximation --
11. Ergodic properties--
Appendices--
References--
Index.

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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