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Abstract: Certain one-dimensional nearest-neighbor random walks in i.i.d. randomspace-time environments are known to have diffusive scaling limits. In thecontinuum limit, the random environment is represented by a `stochastic flow ofkernels-, which is a collection of random kernels that can be looselyinterpreted as the transition probabilities of a Markov process in a randomenvironment. The theory of stochastic flows of kernels was introduced by Le Janand Raimond, who showed that each such flow is characterized by its n-pointmotions. We focus on a class of stochastic flows of kernels with Browniann-point motions which, after their inventors, will be called Howitt-Warrenflows. We give a graphical construction of general Howitt-Warren flows, wherethe underlying random environment takes on the form of a suitably markedBrownian web. Alternatively, we show that a special subclass of theHowitt-Warren flows can be constructed as random flows of mass in a Browniannet. Using these constructions, we prove some new results for the Howitt-Warrenflows. In particular, we show that the kernels spread with a finite speed andhave a locally finite support at deterministic times if and only if the flow isembeddable in a Brownian net. We show that the kernels are always purely atomicat deterministic times, but with the exception of a special subclass known asthe erosion flows, exhibit random times when the kernels are purely non-atomic.We moreover prove ergodic statements for a class of measure-valued processesinduced by the Howitt-Warren flows. Along the way, we also prove some newresults for the Brownian web and net.



Author: Emmanuel Schertzer, Rongfeng Sun, Jan M. Swart

Source: https://arxiv.org/







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