zbMATH — the first resource for mathematics

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 \(\text{const} \cdot e^{- \alpha t}/t^{\nu/2}\).

60G50 Sums of independent random variables; random walks
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
Full Text: Numdam EuDML