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Stochastic flows in the Brownian web and net


About this Title

Emmanuel Schertzer, 109 Montague Street, Brooklyn, New York, New York 11201, Rongfeng Sun, Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076, Singapore and Jan M. Swart, Institute of Information Theory and Automation of the ASCR (ÚTIA), Pod vodárenskou věží 4, 18208 Praha 8, Czech Republic

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 227, Number 1065
ISBNs: 978-0-8218-9088-2 (print); 978-1-4704-1426-9 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2013-00687-9
Published electronically: May 24, 2013
Keywords:Brownian web, Brownian net, stochastic flow of kernels, measure-valued process, Howitt-Warren flow, linear system, random walk in random environment, finite graph representation
MSC: Primary 82C21; Secondary 60K35, 60K37, 60D05

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Table of Contents


Chapters

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

Abstract


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. Our 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.

Our 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'. Our construction for general Howitt-Warren flows is based on a Poisson marking procedure developed by Newman, Ravishankar and Schertzer for the Brownian web. Alternatively, we 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, we prove some new results for the Howitt-Warren flows. In particular, we show that the kernels spread with a finite speed and have a locally finite support at deterministic times if and only if the flow is embeddable in a Brownian net. We show that the kernels are always purely atomic at deterministic times, but, with the exception of the erosion flows, exhibit random times when the kernels are purely non-atomic. We moreover prove ergodic statements for a class of measure-valued processes induced by the Howitt-Warren flows.

Our work also yields some new results in the theory of the Brownian web and net. In particular, we prove several new results about coupled sticky Brownian webs and about a natural coupling of a Brownian web with a Brownian net. We also introduce a `finite graph representation' which gives a precise description of how paths in the Brownian net move between deterministic times.

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