Local equilibrium for a one dimensional zero range process
type: Item
instance of: scholarly article
title: Local equilibrium for a one dimensional zero range process
zbMATH Open document ID: 0626.60105
DOI: 10.1016/0304-4149(87)90049-4
author: Pablo A. Ferrari
author: Errico Presutti
author: Maria Eulalia Vares
published in: Stochastic Processes and their Applications
publication date: 1987-01-01T00:00:00Z
review text: Let \((X(t))_{t\geq 0}\) be the simple zero range process with state space \(N^ Z\) and generator \[ Lf(x)=2^{-1}\sum_{u\in Z}I_{[x_ u>0]}\{f(x^{u,u-1})+f(x^{u,u-1})-2f(x)\},\quad x\in N^ Z, \] \[ where\quad x_ u^{u,v}=x_ u-1,\quad x_ v^{u,v}=x_ v-1,\quad x_ w^{u,v}=x_ w\quad for\quad w\neq u,v. \] Denote by \(T_ t\), \(t\geq 0\), the associated Markov semigroup. Consider the measure \(\mu\) on \(N^ Z\), where \(X_ u(0)\) are independent and geometrically distributed with average \(\rho_-\) at \(u\leq 0\), \(\rho_+\) at \(u>0\) and \(\rho_- >\rho_+>0\). Let \(\rho\) (r,\(\tau)\) \((r\in {\mathbb{R}},\tau >0)\) be the unique solution to the equation \[ \frac{\partial}{\partial \tau}\rho =2^{-1}\frac{\partial}{\partial r}((1+\rho)^{- 2}\frac{\partial}{\partial r}\rho),\quad \lim_{\tau \to 0}\rho (r,\tau)=I_{[r<0]}\rho_-+I_{[r>0]}\rho_+,\quad r\neq 0. \] The authors prove that, for any cylindrical function f on \(N^ Z\) and for any \(\tau >0\) and \(K>0\) \[ \lim_{\epsilon \to 0}\sup_{| \epsilon u| \leq K}| \mu T_{\epsilon^{-2}\tau}(S_ uf)-\mu_{\rho (\epsilon u,\tau)}(f)| =0, \] where \((S_ u,u\in Z)\) is the shift group on \(N^ Z\) and \(\{\mu_{\rho}:\rho >0\}\) denotes the set of the extremal invariant measures of the process.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 4018084
zbMATH Keywords: zero range process
zbMATH Keywords: Markov semigroup
zbMATH Keywords: extremal invariant measures
reviewed by: Mu-Fa Chen
MaRDI profile type: Publication
full work available at URL: 0304-4149(87)90049-4
OpenAlex ID: W1996612102
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description: scientific article; zbMATH DE number 4018084