Stochastic Flows in the Brownian Web and Net
About this ebook
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
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.
