eBook ISBN:  9781470414269 
Product Code:  MEMO/227/1065.E 
List Price:  $86.00 
MAA Member Price:  $77.40 
AMS Member Price:  $51.60 
eBook ISBN:  9781470414269 
Product Code:  MEMO/227/1065.E 
List Price:  $86.00 
MAA Member Price:  $77.40 
AMS Member Price:  $51.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 227; 2014; 160 ppMSC: Primary 82; Secondary 60
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

Chapters

1. Introduction

2. Results for HowittWarren flows

3. Construction of HowittWarren flows in the Brownian web

4. Construction of HowittWarren flows in the Brownian net

5. Outline of the proofs

6. Coupling of the Brownian web and net

7. Construction and convergence of HowittWarren flows

8. Support properties

9. Atomic or nonatomic

10. Infinite starting mass and discrete approximation

11. Ergodic properties

A. The HowittWarren martingale problem

B. The Hausdorff topology

C. Some measurability issues

D. Thinning and Poissonization

E. A onesided version of Kolmogorov’s moment criterion


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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.

Chapters

1. Introduction

2. Results for HowittWarren flows

3. Construction of HowittWarren flows in the Brownian web

4. Construction of HowittWarren flows in the Brownian net

5. Outline of the proofs

6. Coupling of the Brownian web and net

7. Construction and convergence of HowittWarren flows

8. Support properties

9. Atomic or nonatomic

10. Infinite starting mass and discrete approximation

11. Ergodic properties

A. The HowittWarren martingale problem

B. The Hausdorff topology

C. Some measurability issues

D. Thinning and Poissonization

E. A onesided version of Kolmogorov’s moment criterion