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Stochastic Flows in the Brownian Web and Net
Emmanuel Schertzer, Université Pierre et Marie Curie, Paris, France, Rongfeng Sun, National University of Singapore, Singapore, and Jan M. Swart, Academy of Sciences of the Czech Republic, Praha, Czech Republic

Memoirs of the American Mathematical Society
2014; 160 pp; softcover
Volume: 227
ISBN-10: 0-8218-9088-3
ISBN-13: 978-0-8218-9088-2
List Price: US$86
Individual Members: US$51.60
Institutional Members: US$68.80
Order Code: MEMO/227/1065
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It is known that certain one-dimensional nearest-neighbor 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 characterized 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.

Table of Contents

  • Introduction
  • Results for Howitt-Warren flows
  • Construction of Howitt-Warren flows in the Brownian web
  • Construction of Howitt-Warren flows in the Brownian net
  • Outline of the proofs
  • Coupling of the Brownian web and net
  • Construction and convergence of Howitt-Warren flows
  • Support properties
  • Atomic or non-atomic
  • Infinite starting mass and discrete approximation
  • Ergodic properties
  • Appendix A. The Howitt-Warren martingale problem
  • Appendix B. The Hausdorff topology
  • Appendix C. Some measurability issues
  • Appendix D. Thinning and Poissonization
  • Appendix E. A one-sided version of Kolmogorov's moment criterion
  • References
  • Index
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