Local equilibrium for a one dimensional zero range process
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.
scientific article; zbMATH DE number 4018084
scientific article