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October 2002 Characterization of stationary measures for one-dimensional exclusion processes
Maury Bramson, Thomas M. Liggett, Thomas Mountford
Ann. Probab. 30(4): 1539-1575 (October 2002). DOI: 10.1214/aop/1039548366

Abstract

The product Bernoulli measures $\nu_\alpha$ with densities $\alpha$, $\alpha\in [0,1]$, are the extremal translation invariant stationary measures for an exclusion process on $\mathbb{Z}$ with irreducible random walk kernel $p(\cdot)$. Stationary measures that are not translation invariant are known to exist for finite range $p(\cdot)$ with positive mean. These measures have particle densities that tend to 1 as $x\to\infty$ and tend to 0 as $x\to -\infty$; the corresponding extremal measures form a one-parameter family and are translates of one another. Here, we show that for an exclusion process where $p(\cdot)$ is irreducible and has positive mean, there are no other extremal stationary measures. When $\sum_{x<0} x^2 p(x) =\infty$, we show that any nontranslation invariant stationary measure is not a blocking measure; that is, there are always either an infinite number of particles to the left of any site or an infinite number of empty sites to the right of the site. This contrasts with the case where $p(\cdot)$ has finite range and the above stationary measures are all blocking measures. We also present two results on the existence of blocking measures when $p(\cdot)$ has positive mean, and $p(y)\leq p(x)$ and $p(-y)\leq p(-x)$ for $1\leq x\leq y$. When the left tail of $p(\cdot)$ has slightly more than a third moment, stationary blocking measures exist. When $p(-x)\leq p(x)$ for $x>0$ and $\sum_{x<0}x^2p(x)>\infty$, stationary blocking measures also exist.

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Maury Bramson. Thomas M. Liggett. Thomas Mountford. "Characterization of stationary measures for one-dimensional exclusion processes." Ann. Probab. 30 (4) 1539 - 1575, October 2002. https://doi.org/10.1214/aop/1039548366

Information

Published: October 2002
First available in Project Euclid: 10 December 2002

zbMATH: 1039.60086
MathSciNet: MR1944000
Digital Object Identifier: 10.1214/aop/1039548366

Subjects:
Primary: 60K35

Keywords: Blocking measure , Exclusion process , stationary measure

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 4 • October 2002
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