Stochastic Flows in the Brownian Web and Net

Emmanuel Schertzer · Rongfeng Sun · Jan M. Swart

2014年1月 · American Mathematical Soc.

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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
 -point motions. The authors' work focuses on a class of stochastic flows of kernels with Brownian  -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.