  Certain one-dimensional nearest-neighbor random walks in i.i.d. random
space-time environments are known to have diffusive scaling limits. 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 introduced by Le Jan
and Raimond, who showed that each such flow is characterized by its n-point
motions. We focus on a class of stochastic flows of kernels with Brownian
n-point motions which, after their inventors, will be called Howitt-Warren
flows. We give a graphical construction of general Howitt-Warren flows, where
the underlying random environment takes on the form of a suitably marked
Brownian web. Alternatively, we show that a special subclass of the
Howitt-Warren flows can be constructed as random flows of mass in a Brownian
net. Using these constructions, we prove some new results for the Howitt-Warren
flows. In particular, we show that the kernels spread with a finite speed and
have a locally finite support at deterministic times if and only if the flow is
embeddable in a Brownian net. We show that the kernels are always purely atomic
at deterministic times, but with the exception of a special subclass known as
the erosion flows, exhibit random times when the kernels are purely non-atomic.
We moreover prove ergodic statements for a class of measure-valued processes
induced by the Howitt-Warren flows. Along the way, we also prove some new
results for the Brownian web and net.
