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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

Official Date:

2009

Dates:

Date	Event
2009	Published

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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