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The Poisson boundary for homogeneous random walks. (English. Russian original) Zbl 0939.60030
Russ. Math. Surv. 54, No. 2, 441-442 (1999); translation from Usp. Mat. Nauk 54, No. 2, 177-178 (1999).
Let \(\mathcal L\) be a homogeneous Markov chain with a countable space \(X\). \(X^\infty =\{(\alpha _n)_{n=0}^\infty , \alpha _n\in X \}\) is the space of paths of \(\mathcal L\), and \(\mathbf P\) is a probability masure on \(X^\infty \). If \(\mathcal L\) is transient, irreducible and aperiodic, there exists an essentially unique countable decomposition \(\Delta =\{C_0, C_1, \dots\}\) of disjoined sets of positive probability which are invariant w.r.t. shifts \(T^{-1}C=C\), \(T(\alpha _0, \alpha _1,\dots)=(\alpha _1, \alpha _2, \dots)\). Poisson boundary of \(\mathcal L\) is a factor space \((X^\infty , {\mathbf P})/\Delta \). The Poisson boundary is trivial if \(\Delta \) consists of a single set that is an atom. Then, the Poisson boundary for transient, irreducible and aperiodic random walks in \(\mathbb Z^2\) and \(\mathbb Z_+\times \mathbb Z \) is trivial. The structure of Poisson boundary for random walks in \((\mathbb Z_+)^2\) is given in terms of the mean drift vectors. Some generalizations are presented for random walks in \((\mathbb Z_+)^n\) and \((\mathbb Z_+)^n\times \mathbb Z^m\).

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