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Local equilibrium for a one dimensional zero range process. (English) Zbl 0626.60105
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.
Reviewer: Mufa Chen

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 47D07 Markov semigroups and applications to diffusion processes
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