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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. doi:10.1214/08-AOP431
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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: |
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| Volume: | Vol.37 | ||||
| Number: | No.4 | ||||
| Page Range: | pp. 1237-1272 | ||||
| DOI: | 10.1214/08-AOP431 | ||||
| Status: | Peer Reviewed | ||||
| Access rights to Published version: | Restricted or Subscription Access |
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