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July 2009 Consistent families of Brownian motions and stochastic flows of kernels
Chris Howitt, Jon Warren
Ann. Probab. 37(4): 1237-1272 (July 2009). DOI: 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.

Citation

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Chris Howitt. Jon Warren. "Consistent families of Brownian motions and stochastic flows of kernels." Ann. Probab. 37 (4) 1237 - 1272, July 2009. https://doi.org/10.1214/08-AOP431

Information

Published: July 2009
First available in Project Euclid: 21 July 2009

zbMATH: 1203.60123
MathSciNet: MR2546745
Digital Object Identifier: 10.1214/08-AOP431

Subjects:
Primary: 60J60
Secondary: 60K35, 60K35

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.37 • No. 4 • July 2009
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