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Title: Stochastic flows in the Brownian web and net
Authors: Schertzer, E.
Sun, R. 
Swart, J.M.
Keywords: Brownian net
Brownian web
Finite graph representation
Howitt-Warren flow
Linear system
Measurevalued process
Random walk in random environment
Stochastic flow of kernels
Issue Date: Jan-2014
Citation: Schertzer, E., Sun, R., Swart, J.M. (2014-01). Stochastic flows in the Brownian web and net. Memoirs of the American Mathematical Society 227 (1065) : 1-172. ScholarBank@NUS Repository.
Abstract: 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 n-point motions. Our work focuses on a class of stochastic flows of kernels with Brownian n-point motions which, after their inventors, will be called Howitt-Warren flows. Our 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'. Our construction for general Howitt-Warren flows is based on a Poisson marking procedure developed by Newman, Ravishankar and Schertzer for the 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, introduced by Sun and Swart. 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 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. Our work also yields some new results in the theory of the Brownian web and net. In particular, we prove several new results about coupled sticky Brownian webs and about a natural coupling of a Brownian web with a Brownian net. We also introduce a 'finite graph representation' which gives a precise description of how paths in the Brownian net move between deterministic times. © 2013 American Mathematical Society.
Source Title: Memoirs of the American Mathematical Society
ISSN: 00659266
DOI: 10.1090/S0065-9266-2013-00687-9
Appears in Collections:Staff Publications

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