Stochastic flows in the Brownian web and net

0396636 - ÚTIA 2015 RIV US eng J - Journal Article
Schertzer, E. - Sun, R. - Swart, Jan M.
Stochastic flows in the Brownian web and net.
Memoirs of the American Mathematical Society. Roč. 227, č. 1065 (2014), s. 1-160. ISSN 0065-9266. E-ISSN 1947-6221
R&D Projects: GA ČR GA201/07/0237; GA ČR GA201/09/1931
Institutional support: RVO:67985556
Keywords : Brownian web * Brownian net * stochastic flow of kernels * measure-valued process * Howitt-Warren flow * linear system * random walk in random environment * finite graph representation
Subject RIV: BA - General Mathematics
Impact factor: 1.727, year: 2014
http://library.utia.cas.cz/separaty/2013/SI/swart-0396636.pdf

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, ...
Permanent Link: http://hdl.handle.net/11104/0225510