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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. 1237-1272. ISSN 0091-1798

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Official URL: http://dx.doi.org/10.1214/08-AOP431

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 N-dimensional process are distributed as the M-dimensional 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:	0091-1798
Date:	2009
Volume:	Vol.37
Number:	No.4
Page Range:	pp. 1237-1272
Identification Number:	10.1214/08-AOP431
Status:	Peer Reviewed
Access rights to Published version:	Restricted or Subscription Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/37376