Pointwise ergodic theorems for the symmetric exclusion process
type: Item
instance of: scholarly article
title: Pointwise ergodic theorems for the symmetric exclusion process
zbMATH Open document ID: 0621.60114
DOI: 10.1007/BF00320333
publication date: 1987-01-01T00:00:00Z
review text: Let us place one particle in each of some sites of the integer lattice \(S={\mathbb{Z}}^ d\) at time \(t=0\). Each particle will wait an exponentially distributed time of parameter one and then attempt a jump according to a symmetric probability matrix p(x,y). The attempted jump from x to y occurs if and only if y is vacant. Let \(\eta_ t(x)\) be the indicator of the event that \(x\in S\) is occupied at time t \((\eta_ t\) is called the symmetric exclusion process on S). Let \(\mu_{\rho}\), \(0\leq \rho \leq 1\), denote the law according to which each \(x\in S\) is occupied with probability \(\rho\), independently of the others. Suppose that \(\eta_ t\) converges in distribution to \(\mu_{\rho}\) as \(t\to \infty\). Then for all continuous f \[ \lim T^{- 1}\int^{T}_{0}f(\eta_ s)ds=\int fd\mu_{\rho}\quad a.s., \] if only \(d\geq 3\) (Theorem 2.2). The condition \(d\geq 3\) can be dropped when the relative occupation time of a single site is considered, i.e. when \(f(\eta)=\eta (x)\) (Theorem 2.1).
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 4007426
zbMATH Keywords: symmetric exclusion process
zbMATH Keywords: stirring process
zbMATH Keywords: interacting particle systems
zbMATH Keywords: voter model
zbMATH Keywords: ergodic theorem
zbMATH Keywords: integer lattice
zbMATH Keywords: relative occupation time
author: Enrique Daniel Andjel
author: Claude Kipnis
reviewed by: Tamás F. Móri
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4007426