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Interacting random walk in a dynamical random environment. II: Environment from the point of view of the particle. (English) Zbl 0818.60064
Summary: We consider, as in part I (see above), a random walk $$X_ t \in \mathbb{Z}^ \nu$$, $$t \in \mathbb{Z}_ +$$, and a dynamical random field $$\xi_ t (x)$$, $$x \in \mathbb{Z}^ \nu$$, in mutual interaction with each other. The model is a perturbation of an unperturbed model in which walk and field evolve independently. Here we consider the environment process in a frame of reference that moves with the walk, i.e., the “field from the point of view of the particle” $$\eta_ t (\cdot) = \xi_ t (X_ t + \cdot)$$. We prove that its distribution tends, as $$t \to \infty$$, to a limiting distribution $$\mu$$, which is absolutely continuous with respect to the unperturbed equilibrium distribution. We also prove that, for $$\nu \geq 3$$, the time correlations of the field $$\eta_ t$$ decay as $$\text{const} \cdot e^{- \alpha t}/t^{\nu/2}$$.

##### MSC:
 60G50 Sums of independent random variables; random walks 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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