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Consistent families of Brownian motions and stochastic flows of kernels
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Howitt, Chris and Warren, Jon. (2009) Consistent families of Brownian motions and stochastic flows of kernels. The Annals of Probability, Vol.37 (No.4). pp. 12371272. ISSN 00911798

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Official URL: http://dx.doi.org/10.1214/08AOP431
Abstract
Consider the following mechanism for the random evolution of a distribution
of mass on the integer lattice Z. At unit rate, independently for each
site, the mass at the site is split into two parts by choosing a random proportion
distributed according to some specified probability measure on [0, 1] and
dividing the mass in that proportion. One part then moves to each of the two
adjacent sites. This paper considers a continuous analogue of this evolution,
which may be described by means of a stochastic flow of kernels, the theory
of which was developed by Le Jan and Raimond. One of their results is that
such a flow is characterized by specifying its N point motions, which form
a consistent family of Brownian motions. This means for each dimension N
we have a diffusion in RN, whose N coordinates are all Brownian motions.
Any M coordinates taken from the Ndimensional process are distributed as
the Mdimensional process in the family. Moreover, in this setting, the only
interactions between coordinates are local: when coordinates differ in value
they evolve independently of each other. In this paper we explain how such
multidimensional diffusions may be constructed and characterized via martingale
problems.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Statistics 
Library of Congress Subject Headings (LCSH):  Brownian motion processes, Kernel functions, Stochastic processes, Diffusion processes 
Journal or Publication Title:  The Annals of Probability 
Publisher:  Institute of Mathematical Statistics 
ISSN:  00911798 
Official Date:  2009 
Volume:  Vol.37 
Number:  No.4 
Page Range:  pp. 12371272 
Identification Number:  10.1214/08AOP431 
Status:  Peer Reviewed 
Access rights to Published version:  Restricted or Subscription Access 
References:  [1] AMIR, M. (1991). Sticky Brownian motion as the strong limit of a sequence of random walks. 
URI:  http://wrap.warwick.ac.uk/id/eprint/37376 
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