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 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 Not yet published.Expected publication date is January 6, 2014. 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