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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

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 80A20 Heat and mass transfer, heat flow (MSC2010)
##### Keywords:
zero range process; invariant measures; fluctuation fields
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