Memoirs of the American Mathematical Society 2014; 160 pp; softcover Volume: 227 ISBN10: 0821890883 ISBN13: 9780821890882 List Price: US$86 Individual Members: US$51.60 Institutional Members: US$68.80 Order Code: MEMO/227/1065
 It is known that certain onedimensional nearestneighbor random walks in i.i.d. random spacetime 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 HowittWarren flows. The authors' main result gives a graphical construction of general HowittWarren 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 socalled "erosion flow", can be constructed from two coupled "sticky Brownian webs". The authors' construction for general HowittWarren 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 HowittWarren 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 HowittWarren flows. Table of Contents  Introduction
 Results for HowittWarren flows
 Construction of HowittWarren flows in the Brownian web
 Construction of HowittWarren flows in the Brownian net
 Outline of the proofs
 Coupling of the Brownian web and net
 Construction and convergence of HowittWarren flows
 Support properties
 Atomic or nonatomic
 Infinite starting mass and discrete approximation
 Ergodic properties
 Appendix A. The HowittWarren martingale problem
 Appendix B. The Hausdorff topology
 Appendix C. Some measurability issues
 Appendix D. Thinning and Poissonization
 Appendix E. A onesided version of Kolmogorov's moment criterion
 References
 Index
