# Stochastic Flows in the Brownian Web and Net

Share this page
*Emmanuel Schertzer; Rongfeng Sun; Jan M. Swart*

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

# Table of Contents

## Stochastic Flows in the Brownian Web and Net

- Chapter 1. Introduction 18 free
- Chapter 2. Results for Howitt-Warren flows 1118 free
- Chapter 3. Construction of Howitt-Warren flows in the Brownian web 2330
- Chapter 4. Construction of Howitt-Warren flows in the Brownian net 3542
- Chapter 5. Outline of the proofs 4552
- Chapter 6. Coupling of the Brownian web and net 4754
- Chapter 7. Construction and convergence of Howitt-Warren flows 7582
- Chapter 8. Support properties 8794
- Chapter 9. Atomic or non-atomic 97104
- Chapter 10. Infinite starting mass and discrete approximation 111118
- Chapter 11. Ergodic properties 117124
- Appendix A. The Howitt-Warren martingale problem 129136
- Appendix B. The Hausdorff topology 141148
- Appendix C. Some measurability issues 145152
- Appendix D. Thinning and Poissonization 149156
- Appendix E. A one-sided version of Kolmogorov’s moment criterion 151158
- References 153160
- Index 159166 free