Equilibrium fluctuations for zero range processes in random environment
type: Item
instance of: scholarly article
title: Equilibrium fluctuations for zero range processes in random environment
zbMATH Open document ID: 0935.60082
DOI: 10.1016/S0304-4149(98)00044-1
author: A. Koukkous
author: Guy Gielis
author: C. Landim
published in: Stochastic Processes and their Applications
publication date: 1999-11-18T00:00:00Z
review text: A central limit theorem for the density field for stationary zero range processes in a random environment is proved. The evolution of a zero range process in a random environment on a lattice \(Z^d\) is described as follows. If there are \(n\) particles at site \(x\), at rate \( \alpha_x p(y) g(n) \) one of them jumps to site \(x+y\). Here \(\alpha =\{\alpha_x,x\in Z^d\}\) is a family of i.i.d. random variables (random environment), \(p\) is a finite range, symmetric transition probability on \(Z^d\), and \(g:N\to R_+\) with \(0=g(0)<g(k)\) for \(k\geq 1\) is a jump rate. The density field \(Y_t\) is defined as follows: for each compact supported smooth function \(H:R^d\to R\) and each \(t\geq 0\), \[ Y_t^N(H) = N^{-{d/2}} \sum_{x\in Z^d} H(x/N) \left \{\eta_{tN^2}(x) - E_{\overline {\nu}_{\alpha ,\varphi}} [\eta (x)]\right\}, \] where \(\eta_t(x)\) is the number of particles at time \(t\) at site \(x\), \(N^{-1}\) is the scale parameter, and \(\overline {\nu}_{\alpha ,\varphi}\) is an invariant measure (one from a parametric class \(\{\overline {\nu}_{\alpha ,\varphi}\}_{\varphi \geq 0}\)) of the process for the fixed environment \(\alpha \). The main result of the paper states that, averaging over \(\alpha \), the density field converges in distribution to Ornstein-Uhlenbeck process.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B05
zbMATH DE Number: 1364508
zbMATH Keywords: interacting particle system
zbMATH Keywords: hydrodynamic behavior
zbMATH Keywords: central limit theorem
zbMATH Keywords: Boltzmann-Gibbs principle
reviewed by: Martin Janžura
MaRDI profile type: Publication
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full work available at URL: s0304-4149(98)00044-1
OpenAlex ID: W2007931343
description: scientific article; zbMATH DE number 1364508