Stochastic Flows in the Brownian Web and Net (Memoirs of the American Mathematical Society)

[Book Description]

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.

Chapter 1 Introduction                         1  (10)
    Chapter 2 Results for Howitt-Warren flows      11 (12)
    Chapter 3 Construction of Howitt-Warren        23 (12)
    flows in the Brownian web
    Chapter 4 Construction of Howitt-Warren        35 (10)
    flows in the Brownian net
    Chapter 5 Outline of the proofs                45 (2)
    Chapter 6 Coupling of the Brownian web and     47 (28)
    net
    Chapter 7 Construction and convergence of      75 (12)
    Howitt-Warren flows
    Chapter 8 Support properties                   87 (10)
    Chapter 9 Atomic or non-atomic                 97 (14)
    Chapter 10 Infinite starting mass and          111(6)
    discrete approximation
    Chapter 11 Ergodic properties                  117(12)
Appendix A The Howitt-Warren martingale problem    129(12)
Appendix B The Hausdorff topology                  141(4)
Appendix C Some measurability issues               145(4)
Appendix D Thinning and Poissonization             149(2)
Appendix E A one-sided version of Kolmogorov's     151(2)
moment criterion
References                                         153(6)
Index                                              159