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Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces

Published online by Cambridge University Press:  14 June 2011

Y. GUIVARC’H  and
C. R. E. RAJA
Y. GUIVARC’H
Affiliation:
IRMAR, Campus de Beaulieu, Université de Rennes I, 35042 Rennes, France (email: yves.guivarch@univ-rennes1.fr)
C. R. E. RAJA
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560059, India (email: creraja@isibang.ac.in)
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Abstract

We discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular, we show that a closed subgroup of a product of finitely many linear groups over local fields supports an adapted recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces and for associated homeomorphisms with infinite invariant measure. The structural properties of closed subgroups of linear groups over local fields and the properties of group actions with respect to certain Radon measures associated with random walks play an important role in the proofs.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1313 - 1349
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[2]Atkinson, G.. Recurrence of co-cycles and random walks. J. Lond. Math. Soc. (2) 13 (1976), 486–488.CrossRefGoogle Scholar
[3]Baldi, P.. Caractérisation des groupes de Lie connexes récurrents. Ann. Inst. H. Poincaré Sect. B (N.S.) 17 (1981), 281–308.Google Scholar
[4]Babillot, M., Bougerol, P. and Elie, L.. The random difference equation X n=A nX n−1+B n in the critical case. Ann. Probab. 25 (1997), 478–493.CrossRefGoogle Scholar
[5]Baumgartner, U. and Willis, G. A.. Contraction groups and scales of automorphisms of totally disconnected locally compact groups. Israel J. Math. 142 (2004), 221–248.CrossRefGoogle Scholar
[6]Benoist, Y. and Quint, J.-F.. Mesures stationnaires et fermés invariants des espaces homogènes. C. R. Acad. Sci. Paris Sér. I 347 (2009), 9–13.CrossRefGoogle Scholar
[7]Blachère, S., Haissinsky, P. and Mathieu, P.. Harmonic measures versus quasiconformal measures for hyperbolic groups. Preprint, arxiv December 2008.Google Scholar
[8]Bougerol, P. and Lacroix, J.. Products of Random Matrices with Applications to Schrödinger Operators (Progress in Probability and Statistics, 8). Birkhäuser, Boston, MA, 1985.CrossRefGoogle Scholar
[9]Bougerol, P.. Oscillation des produits de matrices aléatoires dont l’exposant de Lyapunov est nul, Lyapunov exponents (Bremen, 1984) (Lecture Notes in Mathematics, 1186). Springer, Berlin, 1986, pp. 27–36.Google Scholar
[10]Cartwright, D. I., Kaimanovich, V. A. and Woess, W.. Random walks on the affine group of local fields and of homogeneous trees. Ann. Inst. Fourier (Grenoble) 44 (1994), 1243–1288.CrossRefGoogle Scholar
[11]Conze, J.-P. and Guivarc’h, Y.. Remarques sur la distalité dans les espaces vectoriels. C. R. Acad. Sci. Paris Sér. A 278 (1974), 1083–1086.Google Scholar
[12]Coudène, Y.. The Hopf argument. J. Mod. Dyn. 1 (2007), 147–153.CrossRefGoogle Scholar
[13]Deroin, B., Kleptsyn, V. and Navas, A.. On the question of ergodicity for minimal group action on the circle. Mosc. Math. J. 9 (2009), 263–303.CrossRefGoogle Scholar
[14]Dudley, R. M.. Random walks on abelian groups. Proc. Amer. Math. Soc. 13 (1962), 447–450.CrossRefGoogle Scholar
[15]Eskin, A. and Margulis, G.. Recurrence properties of random walks on finite volume homogeneous manifolds. Random Walks and Geometry. Walter de Gruyter, Berlin, 2004, pp. 431–444.CrossRefGoogle Scholar
[16]Foguel, S. R.. The Ergodic Theory of Markov Processes (Van Nostrand Mathematical Studies, 21). Van Nostrand Reinhold, New York–Toronto–London, 1969.Google Scholar
[17]Furman, A.. Random Walks on Groups and Random Transformations (Handbook of Dynamical Systems, 1A). North-Holland, Amsterdam, 2002, pp. 931–1014.Google Scholar
[18]Furstenberg, H.. Boundary theory and stochastic processes on homogeneous spaces. Harmonic Analysis on Homogeneous Spaces (Proceedings of Symposia in Pure Mathematics, XXVI, Williams College, Williamstown, MA, 1972). American Mathematical Society, Providence, RI, 1973, pp. 193–229.CrossRefGoogle Scholar
[19]Furstenberg, H.. Noncommuting random products. Trans. Amer. Math. Soc. 108 (1963), 377–428.CrossRefGoogle Scholar
[20]Gallardo, L. and Schott, R.. Marches aléatoires sur les espaces homogènes de certains groupes de type rigide. Conference on Random Walks (Kleebach, 1979) (Astérisque, 74). Société Mathématique, Paris, 1980, pp. 149–170, 4.Google Scholar
[21]Gromov, M.. Groups of polynomial growth and expanding maps. Publ. Math. Inst. Hautes Études Sci. 53 (1981), 53–73.CrossRefGoogle Scholar
[22]Guivarc’h, Y.. Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101 (1973), 333–379.CrossRefGoogle Scholar
[23]Guivarc’h, Y. and Keane, M.. Marches aléatoires transitoires et structure des groupes de Lie (Symposia Mathematica, XXI). Academic Press, London, 1977, pp. 197–217, Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati, INDAM, Rome, 1975.CrossRefGoogle Scholar
[24]Guivarc’h, Y., Keane, M. and Roynette, B.. Marches aléatoires sur les groupes de Lie (Lecture Notes in Mathematics, 624). Springer, Berlin–New York, 1977.CrossRefGoogle Scholar
[25]Guivarc’h, Y.. Quelques propriétés asymptotiques des produits de matrices aléatoires. Eighth Saint Flour Probability Summer School—1978 (Saint Flour, 1978) (Lecture Notes in Mathematics, 774). Springer, Berlin, 1980, pp. 177–250.CrossRefGoogle Scholar
[26]Guivarc’h, Y.. Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés. Ergod. Th. & Dynam. Sys. 9 (1989), 433–453.CrossRefGoogle Scholar
[27]Guivarc’h, Y. and LeJan, Y.. Sur l’enroulement du flot géodésique. C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 645–648.Google Scholar
[28]Guivarc’h, Y.. Marches aléatoires sur les groupes. Development of Mathematics 1950–2000. Birkhäuser, Basel, 2000, pp. 577–608.CrossRefGoogle Scholar
[29]Guivarc’h, Y and Starkov, A. N.. Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms. Ergod. Th. & Dynam. Sys. 24 (2004), 767–802.CrossRefGoogle Scholar
[30]Hennion, H. and Roynette, B.. Un théorème de dichotomie pour une marche aléatoire sur un espace homogène. Conference on Random Walks (Kleebach, 1979) (French) (Astérisque, 74). Société Mathématique, Paris, 1980, pp. 99–122, 4.Google Scholar
[31]Hebisch, W. and Saloff-Coste, L.. Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 (1993), 673–709.CrossRefGoogle Scholar
[32]Jaworski, W. and Raja, C. R. E.. The Choquet–Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth. New York J. Math. 13 (2007), 159–174.Google Scholar
[33]Kesten, H.. The Martin boundary of recurrent random walks on countable groups. Proc. Fifth Berkeley Sympos. Math. Statist. Probab. (Berkeley, CA, 1965/66) (Contributions to Probability Theory, II). University of California Press, Berkeley, CA, 1967, pp. 51–74, Part 2.Google Scholar
[34]Kesten, H.. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973), 207–248.CrossRefGoogle Scholar
[35]Le Page, E.. Théorèmes limites pour les produits de matrices aléatoires. Probability Measures on Groups (Oberwolfach, 1981) (Lecture Notes in Mathematics, 928). Springer, Berlin–New York, 1982,pp. 258–303.CrossRefGoogle Scholar
[36]Lin, M.. Conservative Markov processes on a topological space. Israel J. Math. 8 (1970), 165–186.CrossRefGoogle Scholar
[37]Losert, V.. On the structure of groups with polynomial growth. Math. Z. 195 (1987), 109–117.CrossRefGoogle Scholar
[38]Lyons, T. J. and McKean, H. P.. Winding of the plane Brownian motion. Adv. Math. 51 (1984), 212–225.CrossRefGoogle Scholar
[39]Montgomery, D. and Zippin, L.. Topological Transformation Groups. Interscience, New York–London, 1955.Google Scholar
[40]Margulis, G.. Discrete Subgroups of Semisimple Lie Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17). Springer, Berlin, 1991.CrossRefGoogle Scholar
[41]Moore, C. C.. Ergodicity of flows on homogeneous spaces. Amer. J. Math. 88 (1966), 154–178.CrossRefGoogle Scholar
[42]Parreau, A.. Sous-groupes elliptiques de groupes linéaires sur un corps valué. J. Lie Theory 13 (2003), 271–278.Google Scholar
[43]Peigné, M.. Systèmes d’itérations de transformations aléatoires: propriétés de récurrence, Preprint, Université de Tours, 2008.Google Scholar
[44]Raja, C. R. E.. On classes of p-adic Lie groups. New York J. Math. 5 (1999), 101–105.Google Scholar
[45]Raja, C. R. E.. On growth, recurrence and the Choquet–Deny theorem for p-adic Lie groups. Math. Z. 251 (2005), 827–847.CrossRefGoogle Scholar
[46]Raja, C. R. E. and Schott, R.. Recurrent random walks on homogeneous spaces of p-adic algebraic groups of polynomial growth. Arch. Math. (Basel) 91 (2008), 379–384.CrossRefGoogle Scholar
[47]Rees, M.. Checking ergodicity of some geodesic flows with infinite Gibbs measure. Ergod. Th. & Dynam. Sys. 1 (1981), 107–133.CrossRefGoogle Scholar
[48]Revuz, D.. Markov Chains (North-Holland Mathematical Library, 11). North-Holland–American Elsevier, Amsterdam–Oxford–New York, 1975.Google Scholar
[49]Siebert, E.. Contractive automorphisms on locally compact groups. Math. Z. 191 (1986), 73–90.CrossRefGoogle Scholar
[50]Shalom, Y.. Explicit Kazhdan constants for representations of semisimple and arithmetic groups. Ann. Inst. Fourier 50 (2000), 833–863.CrossRefGoogle Scholar
[51]Stuck, G.. Growth of homogeneous spaces, density of discrete subgroups and Kazhdan’s property (T). Invent. Math. 109(3) (1992), 505–517.CrossRefGoogle Scholar
[52]Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171–202.CrossRefGoogle Scholar
[53]Varopoulos, N. Th., Saloff-Coste, L. and Coulhon, T.. Analysis and Geometry on Groups (Cambridge Tracts in Mathematics, 100). Cambridge University Press, Cambridge, 1992.Google Scholar
[54]Wang, S. P.. The Mautner phenomenon for p-adic Lie groups. Math. Z. 185 (1984), 403–412.CrossRefGoogle Scholar
[55]Woess, W.. Random Walks on Infinite Graphs and Groups (Cambridge Tracts in Mathematics, 138). Cambridge University Press, Cambridge, 2000.CrossRefGoogle Scholar