## The Annals of Probability

### Exclusion processes in higher dimensions: Stationary measures and convergence

#### Abstract

There has been significant progress recently in our understanding of the stationary measures of the exclusion process on Z. The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on Zd and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on Zd. In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as t→∞ for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence.

#### Article information

Source
Ann. Probab., Volume 33, Number 6 (2005), 2255-2313.

Dates
First available in Project Euclid: 7 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1133965859

Digital Object Identifier
doi:10.1214/009117905000000341

Mathematical Reviews number (MathSciNet)
MR2184097

Zentralblatt MATH identifier
1099.60067

#### Citation

Bramson, M.; Liggett, T. M. Exclusion processes in higher dimensions: Stationary measures and convergence. Ann. Probab. 33 (2005), no. 6, 2255--2313. doi:10.1214/009117905000000341. https://projecteuclid.org/euclid.aop/1133965859

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