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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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##### References:
 [1] Andjel, E., Invariant measures for the zero range process, Ann. of prob., 10, 525-547, (1982) · Zbl 0492.60096 [2] Arratia, R., The motion of a tagged particle in the simple symmetric exclusion system on Z, Ann. of prob., 11, 362-373, (1983) · Zbl 0515.60097 [3] Dawson, D., Critical dynamics and fluctuations for a Mean field model of cooperative behavior, J. stat. phys., 31, 29-85, (1983) [4] De Masi, A.; Ianiro, N.; Pellegrinotti, A.; Presutti, E., A survey of the hydrodynamical behavior of many particle systems, () · Zbl 0567.76006 [5] Ferrari, P.A.; Presutti, E.; Scacciatelli, E.; Vares, M.E., The symmetric simple exclusion process, I: probability estimates, (1986), Preprint · Zbl 0749.60094 [6] Galves, A.; Kipnis, C.; Marchioro, C.; Presutti, E., On equilibrium measures which exhibit a temperature gradient: study of a model, Comm. math. phys., 81, 127-147, (1981) · Zbl 0465.60089 [7] Liggett, T.M., Interacting particle systems, (1985), Springer-Verlag Berlin · Zbl 0559.60078 [8] McKean, H.P., Propagation of chaos for a class of nonlinear parabolic equations, (), 41-57, Catholic University [9] Morrey, C.B., On the derivation of the equation of hydrodynamics from statistical mechanics, Comm. pure appl. math., 8, 279-326, (1955) · Zbl 0065.19406 [10] Oelschlager, K., A law of large numbers for moderately interacting diffusion processes, Z. wahrsch. verw. geb., 69, 279-322, (1985) · Zbl 0549.60071 [11] Presutti, E., Collective phenomena in stochastic particle systems, () · Zbl 0637.76058 [12] Presutti, E.; Spohn, H., Hydrodynamics of the voter model, Ann. prob., 11, 867-875, (1983) · Zbl 0527.60094 [13] Rokhlin, V.A., On the fundamental ideas of measure theory, Amer. math. soc. transl., 10, 1-54, (1962) [14] Rost, H., Nonequilibrium behavior of a many particle system: density profile and local equilibrium, Z. wahrsch. verw. geb., 58, 41-53, (1981) · Zbl 0451.60097 [15] Shiga, T.; Tanaka, H., Central limit theorem for a system of markovian particles with Mean field interactions, Z. wahrsch. verw. geb., 69, 439-459, (1985) · Zbl 0607.60095 [16] Spohn, H., Kinetic equations for Hamiltonian systems: Markovian limits, Rev. mod. phys., 53, 569-615, (1980) [17] Spohn, H., Hydrodynamic limit for systems with many particles: A bibliography, J. stat. phys., 44, 1064-1067, (1986) [18] Tanaka, H.; Hitsuda, M., Central limit theorems for a simple diffusion model of interacting particles, Hiroshima math. J., 11, 415-423, (1981) · Zbl 0469.60097 [19] Rost, H., Diffusion de spheres dures dans la droite réelle: comportement macroscopique et equilibre local, (), 127-143 · Zbl 0544.60098
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