Stochastic flows in the Brownian web and net

TL;DR

This paper introduces graphical and net-based constructions of Howitt-Warren stochastic flows with Brownian n-point motions, revealing their spreading properties, atomicity, and ergodic behavior, and extends understanding of the Brownian web and net.

Contribution

It provides new constructions and results for Howitt-Warren flows, linking them to Brownian webs and nets, and characterizes their support, atomicity, and ergodic properties.

Findings

Kernels spread with finite speed

Kernels have locally finite support at deterministic times

Kernels are purely atomic at deterministic times

Abstract

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.