Characterization of stationary measures for one-dimensional exclusion processes.
type: Item
instance of: scholarly article
title: Characterization of stationary measures for one-dimensional exclusion processes.
zbMATH Open document ID: 1039.60086
author: Maury Bramson
author: Thomas M. Liggett
author: Thomas Mountford
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The authors provide a characterization of stationary measures for one-dimensional exclusion processes in adding to known results their new results as follows. The one-dimensional exclusion process with random walk kernel is a continuous time Markov process on \(\{0,1\}^{\mathbb{Z}} \). Particles can only jump on empty sites, and this jump probability is given by the random walk kernel \( p \). If the random walk kernel is irreducible, it is well known that the Bernoulli product measures are the extremal translation invariant stationary measures for the process. When the mean \(\mu:=\sum_{x}xp(x) =0 \), there are no non-translation invariant stationary measures. Stationary measures that are not translation invariant are known to exist for finite range \( p \) with positive mean (i.e. there is a bias to the right). These so-called profile/blocking measures have particle densities that tend to \( 1 \) as \( x\to\infty \) and tend to \( 0 \) as \( x\to -\infty \). The first main result of the article goes in the opposite direction, i.e., the authors show that, when \( p \) is irreducible with positive mean, the only possible extremal non translation invariant stationary measures consist of a profile measure together with its translates. The second result shows that for \(p \) with finite mean but \(\sum_{x<0}x^2p(x)=\infty \) any non translation invariant stationary measure is not a blocking measure. The authors also present two results on the existence of blocking measures under the monotonicity condition on \(p \). In particular they substantially relax the tail behaviour assumed by \textit{M. Bramson} and \textit{T. Mountford} [Ann. Probab. 30, 1082--1130 (2002; Zbl 1042.60062)] when they assume slightly more than three moments on the left tail. In this case stationary blocking measures do exist. The last result gives the existence of a stationary blocking measure when \(p \) has finite strict positive mean with \( p(y)\leq p(x) \) and \( p(-y)\leq p(-x) \) for \( 1\…
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 82C22
zbMATH DE Number: 1906094
zbMATH Keywords: exclusion process
zbMATH Keywords: stationary measure
zbMATH Keywords: blocking measure
reviewed by: Stefan Adams
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1039548366
OpenAlex ID: W2021348160
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DOI: 10.1214/AOP/1039548366
description: scientific article; zbMATH DE number 1906094