Boundaries and harmonic functions for random walks with random transition probabilities

Journal of Theoretical Probability

Volumn 17, Issue 3, 2004, Pages 605-646

  Kaimanovich, V A ^a   Kifer, Y ^b   Rubshtein, B Z ^c  

^a UNIVERSITÉ DE RENNES 1   (France)

^b HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^c BEN GURION UNIVERSITY OF THE NEGEV   (Israel)

Author keywords

harmonic function; Poisson boundary; random transition probability; Random walk

Indexed keywords

EID: 4544348714     PISSN: 08949840     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:JOTP.0000040291.80182.65     Document Type: Article

References (36)

1. Bridson, M. R., and Haefliger, A. (1999). Metric spaces of Non-Positive Curvature, Springer-Verlag New York.
2. Cornfeld, I. P., Fomin, S. V., and Sinai Ya. G., (1982). Ergodic Theory Springer-Verlag New York.
3. Cartwright, D. I., and Soardi, P. M., (1989). Convergence to ends for random walks on the automorphism group of a tree. Proc. Amer. Math. Soc. 107, 817-823.
4. Derriennic, Y. (1976). Lois "zéro ou deux" pour les processus de Markov, applications aux marches aléatoires. Ann. Inst. H. Poincaré, Sect. B 12, 111-129.
5. Derriennic, Y. (1980). Quelques applications du théorème ergodique sous-additif. Astérisque 74, 183-201.
6. Derriennic, Y. (1986). Entropie, théorèmes limites et marches aléatoires. Springer Lecture Notes in Math. 1210, 241-284.
7. Feldman J., and Moore, C. C. (1977). Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Am. Math. Soc. 234, 289-324.
8. Furstenberg, H. (1963). A Poisson formula for semi-simple Lie groups. Ann. Math. 77, 335-386.
9. Furstenberg, H. (1971). Random Walks and Discrete Subgroups of Lie Groups. Adv. Probab. Relat. Top. vol. 1, Dekker New York, 3-63.
10. Greschonig, G. and Schmidt, K. (2000). Ergodic decomposition of quasi-invariant probability measures. Colloq. Math. 84/85, 495-514.
11. Guivarc'h, Y. (1980). Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire Astérisque 74, 47-98.
12. Kaimanovich, V. A. (1991). Poisson boundaries of random walks on discrete solvable groups. In Heyer, H. (ed.), (Proceedings of Conference on Probability Measures on Groups X, (1990), Oberwolfach), Plenum, New York, pp. 205-238.
13. Kaimanovich, V. A. (1992). Measure-theoretic boundaries of Markov chains. In Picardello, M. A. (ed.), 0-2 Laws and Entropy, (Proceedings of the Conference on Harmonic Analysis and Discrete Potential Theory, 1991, Frascati), Plenum, New York, pp. 145-180.
14. Kaimanovich, V. A. (1995). The Poisson boundary of covering Markov operators. Israel J. Math. 89, 77-134.
15. d-Actions (Proceedings of the Warwick Symposium 1993-4), Pollicott, M. and Schmidt, K. (eds.), London Mathematics Society Lecture Note Series, Vol. 228, Cambridge Univ. Press, Cambridge, pp. 127-176.
16. Kaimanovich, V. A. (2000). The Poisson formula for groups with hyperbolic properties. Ann. Math. 152, 659-692.
17. Kalikow, S. (1981). Generalized random walk in a random environment. Ann. Prob. 9, 753-768.
18. Kifer, Y. (1986). Ergodic Theory of Random Transformations, Birkhäuser, Boston.
19. Kifer, Y. (1996). Perron-Frobenius theorem, large deviations, and random perturbations in random environments. Math. Zeit 222, 677-698.
20. Kifer, Y. (2001). "Random" random matrix products. J. Anal. Math. 83, 41-88.
21. Kozlov, S. M., and Molchanov, S. A. (1984). Conditions for the applicability of the central limit theorem to random walks on a lattice. Soviet Math. Dokl. 30, 410-413.
22. Karlsson, A., Margulis, G. A. (1999). A multiplicative ergodic theorem and nonpositively curved spaces. Comm. Math. Phys. 208, 107-123.
23. Krengel, U. (1985). Ergodic Theorems, de Gruyter, Berlin.
24. Kramli, A., and Szasz, D. (1983). Random walks with internal degrees of freedom. I. Local limit theorems. Z. Wahrsch. Verw. Gebiete 63, 85-95.
25. Kaloshin, V. Yu., and Sinai, Ya. G. (2000). Simple random walks along orbits of Anosov diffeomorphisms. Tr. Mat. Inst. Steklova 228, 236-245.
26. Kaimanovich, V. A., and Vershik, A. M. (1983). Random walks on discrete groups: Boundary and entropy. Ann. Probab. 11, 457-490.
27. Lin, M., Rubshtein, B.-Z., and Wittmann R. (1994). Limit theorems for random walks with dynamical random transitions. Probab. Th. Rel. Fields 100, 285-300.
28. Mindlin, D. S., and Rubshtein, B.-Z. (1988). Convolution of random measures on a compact group. Th. Probab. Appl. 33, 355-357.
29. Mindlin, D. S., and Rubshtein, B.-Z. (1994). Convolutional attractors of stationary sequences of random measures on compact groups. Ann. Inst. H. Poincaré 30, 213-233.
30. Orey, S. (1991). Markov chains with stochastically stationary transition probabilities. Ann. Probab. 19, 907-928.
31. Revuz, D. (1984). Markov Chains, 2nd ed., North-Holland, Amsterdam.
32. Rokhlin, V. A. (1949/1952). On the fundamental ideas of measure theory. Mat. Sbornik N. S. 25(67), 107-150 (Russian); English Transl. in Am. Math. Soc. Translations 71, 1-55.
33. Rokhlin, V. A. (1949/1952). On the fundamental ideas of measure theory. Mat. Sbornik N. S. 25(67), 107-150 (Russian); English Transl. in Am. Math. Soc. Translations 71, 1-55.
34. Rokhlin, V. A. (1967). Lectures on the entropy theory of measure preserving transformations. Russian Math. Surveys 22(5), 1-52.
35. Rubshtein, B.-Z. (1995). Convolutions of random measures on compact groups. J. Theor. Probab. 8, 523-538.
36. Woess, W. (1993). Fixed sets and free subgroups of groups acting on metric spaces. Math. Zeit. 214, 425-440.