Asymptotic behavior of a tagged particle in the exclusion process on parallel lattices

We investigate a tagged particle in the exclusion processes on {1, ..., N}×ℤd, with different densities in different levels {k}×ℤd, ∀ k. Ignoring the level the tagged particle lying in, we only concern its position in ℤd, denoted by Xt. Note that the whole space is not homogeneous. We define the environment process viewed from the tagged particle, of which Xt can be expressed as a functional. It is called the tagged particle process. We show the ergodicity of the tagged particle process, then prove the strong law of large numbers. Furthermore, we show the central limit theorem of Xt provided the zero-mean condition.