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On some random walks on $$\mathbb{Z}$$ in random medium. (English) Zbl 1021.60034
The following model of random walks in random environment is considered. Let $$(\Omega, {\mathcal F}, \mu, T)$$ be an invertable dynamical system, i.e., $$(\Omega, {\mathcal F}, \mu)$$ is a probability space and $$T$$ is a map on $$\Omega$$, invertable and measurable together with its inverse. It is ergodic with respect to $$\mu$$ and measure preserving. Given integers $$L, R \geq 1$$, the set $$\Lambda$$ is defined as $$\Lambda = \{ -L, -L+1 , \dots , 0 , \dots , R\}$$. This is the set of possible jumps of the random walk on $$\mathbb Z$$. The probabilities of such jumps are given by a collection $$(p_z)_{z \in \Lambda}$$ of positive random variables on $$(\Omega, {\mathcal F}, \mu)$$. They are supposed to obey the conditions: $$\sum_{z \in \Lambda }p_z (\omega) = 1$$ $$\mu$$-almost surely; $$\exists \varepsilon >0$$ such that $$\forall z \in \Lambda \setminus \{0\}$$, one has $$(p_z /p_R) \geq \varepsilon$$ $$\mu$$-almost surely. The random walk $$\xi_n (\omega)$$ is defined as a Markov chain by the transition probabilities ${\mathcal P}_\omega (\xi_{n+1} (\omega) = x + z \mid \xi_n (\omega) = x) = p_z (T^x \omega), \quad \;x \in {\mathbb Z}, \quad z \in \Lambda,$ and $$\xi_0 (\omega) = 0$$. For this model with $$R=1$$ the following results are obtained. A recurrence criterion expressed in terms of the sign of the maximal Lyapunov exponent of a certain random matrix is provided. An algorithm of calculating of this exponent is given. The existence of the absolutely continuous invariant measure for the Markov chain of “environments viewed from the particle” and of the nonzero drift (in the transient case) are characterized. A sufficient condition for the validity of the central limit theorem in the transient case is given.

MSC:
 60G50 Sums of independent random variables; random walks 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics 60K37 Processes in random environments 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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