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Equilibrium fluctuations of stochastic particle systems: The role of conserved quantities. (English) Zbl 0546.60098
Let \((X(t))_{t\geq 0}\) be an attractive zero range process with operator: \(Lf(x)=\sum_{j\in {\mathbb{Z}}^ d}c(x_ j)\sum_{| k- j| =1}(f(x^{jk})-f(x))\). Its semigroup \(\{T_ t\}_{t\geq 0}\) has the invariant measures described by some extremals \(\mu_ u\), where u is a parameter. Fix u and set \(\mu =\mu_ u\), \(\rho ={\mathbb{E}}X_ i(t)\), \(i\in {\mathbb{Z}}^ d\). Define \(f_ 0(x)=x_ 0-\rho\), \(E_ 0(f,g)=\sum_{j}\int fS_ jgd\mu\), and \(E_ t(f,g)=\sum_{j}\int fS_ jT_ tgd\mu\) where \(S_ kf\) means the function shifted by a vector \(k\in {\mathbb{Z}}^ d.\)
In this paper, the main result is \[ \lim_{t\to\infty }E_ t(f,g)=E_ 0(g,f_ 0)\int (x_ 0-\rho)^ 2d\mu /E_ 0(f,f_ 0) \] for all f,\(g\in {\mathcal D}\) which is crudely the set of all cylindrical functions with mean zero. The second result is \[ \lim_{\epsilon\to 0}\hat A(\epsilon\theta,s\epsilon^{-2})=\exp [-{1\over2}\lambda s|\theta |^ 2]\int (x_ 0-\rho)^ 2d\mu \] for fixed \(\theta\in {\mathbb{R}}^ d\) and s, where \(\lambda =2u/\int (x_ 0-\rho)^ 2d\mu\) and \(\hat A(\theta\),t)\(=\sum_{j}e^{ij\cdot\theta }{\mathbb{E}}[(X_ 0(0)-\rho)(X_ j(t)-\rho)]\). Finally, the last result is translated into the language of fluctuation fields.
Reviewer: M.Chen

60K35 Interacting random processes; statistical mechanics type models; percolation theory
80A20 Heat and mass transfer, heat flow (MSC2010)
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