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Journal Article

Stochastic flows in the Brownian web and net

Schertzer E., Sun R., Swart Jan M.

: Memoirs of the American Mathematical Society vol.227, 1065 (2014), p. 1-160

: GA201/07/0237, GA ČR, GA201/09/1931, GA ČR

: Brownian web, Brownian net, stochastic flow of kernels, measure-valued process, Howitt-Warren flow, linear system, random walk in random environment, finite graph representation

: 10.1090/S0065-9266-2013-00687-9


(eng): 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, ...

: BA