Brémont, Julien One-dimensional finite range random walk in random medium and invariant measure equation. (English) Zbl 1171.60395 Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 1, 70-103 (2009). Summary: We consider a model of random walks on \(\mathbb Z\) with finite range in a stationary and ergodic random environment. We first provide a fine analysis of the geometrical properties of the central left and right Lyapunov eigenvectors of the random matrix naturally associated with the random walk, highlighting the mechanism of the model. This allows us to formulate a criterion for the existence of the absolutely continuous invariant measure for the environments seen from the particle. We then deduce a characterization of the non-zero-speed regime of the model. Cited in 10 Documents MSC: 60K37 Processes in random environments 60F15 Strong limit theorems 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) Keywords:finite range Markov chain; Lyapunov eigenvector; invariant measure; stable cone × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] S. Alili. Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36 (1999) 334-349. · Zbl 0946.60046 · doi:10.1239/jap/1032374457 [2] L. Arnold. Random Dynamical Systems . Springer, Berlin, 1998. · Zbl 0906.34001 [3] E. Bolthausen and I. Goldsheid. Recurrence and transience of random walks in random environments on a strip. Comm. Math. Phys. 214 (2000) 429-447. · Zbl 0985.60092 · doi:10.1007/s002200000279 [4] E. Bolthausen and I. Goldsheid. Lingering random walks in random environment on a strip. Comm. Math. Phys. 278 (2008) 253-288. · Zbl 1142.82007 · doi:10.1007/s00220-007-0390-4 [5] J. Brémont. 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