The invariant measures for a Markovian operator corresponding to a random walk, in a random stationary one-dimensional environment defined by a dynamical system, are quasi-invariant measures for the system. We discuss the construction of such measures in the general case and show unicity, under some assumptions, for a rotation on the circle.
@article{bwmeta1.element.bwnjournal-article-cmv84i2p457bwm,
author = {Jean-Pierre Conze and Yves Guivarc'h},
title = {Marches en milieu al\'eatoire et mesures quasi-invariantes pour un syst\`eme dynamique},
journal = {Colloquium Mathematicae},
volume = {84/85},
year = {2000},
pages = {457-480},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv84i2p457bwm}
}
Conze, Jean-Pierre; Guivarc'h, Yves. Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique. Colloquium Mathematicae, Tome 84/85 (2000) pp. 457-480. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv84i2p457bwm/
[000] [1] S. Alili, Processus de branchement et marche aléatoire en milieux désordonnés, thèse, Université Pierre et Marie Curie (Paris VI), 1993.
[001] [2] D. Anosov, On the additive functional homological equation associated with an irrational rotation of the circle, Izv. Akad. Nauk SSSR 37 (1973), 1259-1274 (in Russian). | Zbl 0298.28016
[002] [3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer, 1975. | Zbl 0308.28010
[003] [4] J. Brémont, Comportement des sommes ergodiques pour des rotations et des fonctions continues peu régulières, Publications des Séminaires de Rennes, 1999.
[004] [5] J.-P. Conze, Equirépartition et ergodicité de transformations cylindriques, Publications des Séminaires de Rennes, 1976.
[005] [6] J.-P. Conze et Y. Guivarc'h, Croissance des sommes ergodiques et principe va- riationnel, preprint, Rennes, 1997.
[006] [7] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Math. 527, Springer, 1976. | Zbl 0328.28008
[007] [8] H. Federer, Geometric Measure Theory, Classics in Math., Springer, 1996. | Zbl 0874.49001
[008] [9] Y. Guivarc'h et J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Ann. Inst. H. Poincaré Sér. Probab. Statist. 24 (1988), 73-98. | Zbl 0649.60041
[009] [10] G. Halász, Remarks on the remainder in Birkhoff's ergodic theorem, Acta Math. Acad. Sci. Hungar. 28 (1976), 389-395. | Zbl 0336.28005
[010] [11] M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. Inst. Hautes Etudes Sci. 49 (1979), 5-233. | Zbl 0448.58019
[011] [12] H. Kesten, M. V. Kozlov and F. Spitzer, A limit law for random walks in a random environment, Compositio Math. 30 (1975), 145-168. | Zbl 0388.60069
[012] [13] S. M. Kozlov, The method of averaging and walks in inhomogeneous environments, Russian Math. Surveys 40 (1985), no. 2, 73-145. | Zbl 0615.60063
[013] [14] S. M. Kozlov and S. A. Molchanov, On conditions for applicability of the central limit theorem to random walks on a lattice, Soviet Math. Dokl. 30 (1984), 410-413. | Zbl 0603.60020
[014] [15] M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Noordhoff, Gro- ningen, 1964.
[015] [16] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985.
[016] [17] A. V. Letchikov, A criterion for applicability of the CLT to one-dimensional random walks in random environments, Theory Probab. Appl. 37 (1992), 553-557. | Zbl 0787.60088
[017] [18] W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb. 25, Springer, 1993. | Zbl 0791.58003
[018] [19] S. A. Molchanov, Lectures on random media, in: Lectures on Probability Theory (Saint-Flour, 1992), Lecture Notes in Math. 1581, Springer, 1994, 242-411. | Zbl 0814.60093
[019] [20] M. F. Norman, Markov Processes and Learning Models, Academic Press, New York, 1972. | Zbl 0262.92003
[020] [21] Y. Peres, A combinatorial application of the maximal ergodic theorem, Bull. London Math. Soc. 20 (1988), 248-252. | Zbl 0642.10051
[021] [22] Ya. G. Sinai, Construction of Markov partitions, Funktsional. Anal. i Prilozhen. 2 (1968), no. 3, 70-80 (in Russian).
[022] [23] Ya. G. Sinai, The limiting behaviour of a one-dimensional random walk in a random environment, Theory Probab. Appl. 27 (1982), 256-268.
[023] [24] Ya. G. Sinai, Simple random walks on tori, preprint.
[024] [25] R. Sine, On invariant probabilities for random rotations, Israel J. Math. 33 (1979), 384-388. | Zbl 0435.60075
[025] [26] F. Solomon, Random walks in a random environment, Ann. Probab. 3 (1975), 1-31. | Zbl 0305.60029