J. A. López and A. Candel, Generic aspects of the geometry of leaves of foliations, 2003.

M. Arnaudon, B. K. Driver, and A. Thalmaier, Gradient estimates for positive harmonic functions by stochastic analysis, Stochastic Processes and their Applications, vol.117, issue.2, pp.202-220, 2007.
DOI : 10.1016/j.spa.2006.07.002

]. A. Anc90 and . Ancona, Théorie du potentiel sur les graphes et les variétés, École d'été de Probabilités de Saint-Flour XVIII?1988, pp.1-112, 1990.

T. Michael, R. Anderson, and . Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math, vol.121, issue.23, pp.429-461, 1985.

M. Arnaudon and A. Thalmaier, Brownian motion and negative curvature In Random walks, boundaries and spectra, Progr. Probab, vol.64, pp.143-161, 2011.

A. Avez, Théorème de Choquet-Deny pour les groupes à croissance non exponentielle, C. R. Acad. Sci. Paris Sér. A, vol.279, pp.25-28, 1974.

]. A. Ave76 and . Avez, Harmonic functions on groups, Differential geometry and relativity, pp.27-32, 1976.

D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol.33, 2001.
DOI : 10.1090/gsm/033

I. Benjamini and N. Curien, Ergodic theory on stationary random graphs, Electronic Journal of Probability, vol.17, issue.0, pp.1-20, 2012.
DOI : 10.1214/EJP.v17-2401

P. Billingsley, Convergence of probability measures Wiley Series in Probability and Statistics: Probability and Statistics, 1999.

L. Brewin, A brief introduction to Cadabra: A tool for tensor computations in General Relativity, Computer Physics Communications, vol.181, issue.3, pp.489-498, 2010.
DOI : 10.1016/j.cpc.2009.10.020

S. Brofferio, M. Salvatori, and W. Woess, Brownian Motion and Harmonic Functions on Sol(p,q), International Mathematics Research Notices, issue.22, pp.5182-5218, 2012.
DOI : 10.1093/imrn/rnr232

URL : https://hal.archives-ouvertes.fr/hal-00710645

A. Candel, The harmonic measures of Lucy Garnett, Advances in Mathematics, vol.176, issue.2, pp.187-247, 2003.
DOI : 10.1016/S0001-8708(02)00036-1

I. Chavel, Eigenvalues in Riemannian geometry Including a chapter, of Pure and Applied Mathematics, 1984.

J. Cheeger, Finiteness Theorems for Riemannian Manifolds, American Journal of Mathematics, vol.92, issue.1, pp.61-74, 1970.
DOI : 10.2307/2373498

[. Camacho and A. Neto, Geometric theory of foliations
DOI : 10.1007/978-1-4612-5292-4

J. Cristina, Gromov-hausdorff convergence of metric spaces, 2008.

Y. Derriennic and . Lois, zéro ou deux " pour les processus de Markov Applications aux marches aléatoires, Ann. Inst. H. Poincaré Sect. B (N.S.), vol.12, issue.2, pp.111-129, 1976.

B. [. Debiard, E. Gaveau, and . Mazet, Th??or??mes de comparaison en g??om??trie riemannienne, Publications of the Research Institute for Mathematical Sciences, vol.12, issue.2, pp.391-42577, 1976.
DOI : 10.2977/prims/1195190722

A. Del and J. , On the decomposition of a subadditive stochastic process, Ann. Probability, vol.5, issue.2, pp.298-302, 1977.

J. J. Duistermaat and J. A. Kolk, Lie groups. Universitext, 2000.

B. Deroin and V. Kleptsyn, Random Conformal Dynamical Systems, Geometric and Functional Analysis, vol.17, issue.4, pp.1043-1105, 2007.
DOI : 10.1007/s00039-007-0606-y

URL : https://hal.archives-ouvertes.fr/hal-00463764

. Thierry-de-la-rue, Espaces de Lebesgue, Séminaire de Probabilités, XXVII, pp.15-21, 1993.
DOI : 10.1007/BFb0087958

E. B. Davies and N. Mandouvalos, Heat Kernel Bounds on Hyperbolic Space and Kleinian Groups, Proc. London Math. Soc. (3), pp.182-208, 1988.
DOI : 10.1112/plms/s3-57.1.182

L. Joseph and . Doob, Classical potential theory and its probabilistic counterpart, Classics in Mathematics, 2001.

E. Paul and . Ehrlich, Continuity properties of the injectivity radius function, Compositio Math, vol.29, pp.151-178, 1974.

J. Eichhorn, The Boundedness of Connection Coefficients and their Derivatives, Mathematische Nachrichten, vol.19, issue.1, pp.145-158, 1991.
DOI : 10.1002/mana.19911520113

R. Edwards, K. Millett, and D. Sullivan, Foliations with all leaves compact, Topology, vol.16, issue.1, pp.13-32, 1977.
DOI : 10.1016/0040-9383(77)90028-3

K. [. Epstein, D. Millett, and . Tischler, Leaves Without Holonomy, Journal of the London Mathematical Society, vol.2, issue.3, pp.16548-552, 1977.
DOI : 10.1112/jlms/s2-16.3.548

]. D. Eps72 and . Epstein, Periodic flows on three-manifolds, Ann. of Math, vol.95, issue.2, pp.66-82, 1972.

]. D. Eps76 and . Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble ), vol.26, issue.1, pp.265-282, 1976.

E. [. Epstein and . Vogt, A Counterexample to the Periodic Orbit Conjecture in Codimension 3, The Annals of Mathematics, vol.108, issue.3, pp.539-552, 1978.
DOI : 10.2307/1971187

M. Einsiedler and T. Ward, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, vol.259, 2011.

H. Furstenberg, Noncommuting random products, Transactions of the American Mathematical Society, vol.108, issue.3, pp.377-428, 1963.
DOI : 10.1090/S0002-9947-1963-0163345-0

K. Peter, N. B. Friz, and . Victoir, Multidimensional stochastic processes as rough paths, Theory and applications, 2010.

L. Garnett, Foliations, the ergodic theorem and Brownian motion, Journal of Functional Analysis, vol.51, issue.3, pp.285-311, 1983.
DOI : 10.1016/0022-1236(83)90015-0

É. Ghys, Topologie des Feuilles Generiques, The Annals of Mathematics, vol.141, issue.2, pp.387-422, 1995.
DOI : 10.2307/2118526

F. [. Gouëzel, F. Mathéus, and . Maucourant, Sharp lower bounds for the asymptotic entropy of symmetric random walks. ArXiv e-prints, 2012.

R. M. Gray, Entropy and information theory, 2011.

A. Grigor-'yan, Heat kernel and analysis on manifolds, AMSIP Studies in Advanced Mathematics, vol.47, 2009.

M. Gromov, Groups of polynomial growth and expanding maps, Publications math??matiques de l'IH??S, vol.2, issue.1, pp.53-73, 1981.
DOI : 10.1007/BF02698687

A. Hatcher, Algebraic topology, 2002.

E. Heintze and H. Hof, Geometry of horospheres, Journal of Differential Geometry, vol.12, issue.4, pp.481-491, 1977.
DOI : 10.4310/jdg/1214434219

E. P. Hsu, Stochastic analysis on manifolds, Graduate Studies in Mathematics, vol.38, 2002.
DOI : 10.1090/gsm/038

K. Ichihara, Comparison theorems for Brownian motions on Riemannian manifolds and their applications, Journal of Multivariate Analysis, vol.24, issue.2, pp.177-188, 1988.
DOI : 10.1016/0047-259X(88)90034-6

]. V. Ka?-i86 and . Ka?-imanovich, Brownian motion and harmonic functions on covering manifolds . An entropic approach, Dokl. Akad. Nauk SSSR, vol.288, issue.5, pp.1045-1049, 1986.

]. V. Ka?-i88 and . Ka?-imanovich, Brownian motion on foliations: entropy, invariant measures, mixing, Funktsional. Anal. i Prilozhen, vol.22, issue.4, pp.82-83, 1988.

A. Vadim and . Kaimanovich, Measure-theoretic boundaries of Markov chains, 0-2 laws and entropy, Harmonic analysis and discrete potential theory, pp.145-180, 1991.

A. S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol.156, 1995.
DOI : 10.1007/978-1-4612-4190-4

J. F. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B, vol.30, pp.499-510, 1968.

J. F. Kingman, Subadditive Ergodic Theory, M. Hammersley, and a reply by the author, pp.883-909, 1973.
DOI : 10.1214/aop/1176996798

URL : http://projecteuclid.org/download/pdf_1/euclid.aop/1176996798

A. Karlsson and F. Ledrappier, Propri??t?? de Liouville et vitesse de fuite du mouvement brownien, Comptes Rendus Mathematique, vol.344, issue.11, pp.685-690, 2007.
DOI : 10.1016/j.crma.2007.04.019

A. Karlsson and F. Ledrappier, Noncommutative ergodic theorems, Geometry, rigidity, and group actions, pp.396-418
URL : https://hal.archives-ouvertes.fr/hal-00607743

A. [. Ka?-imanovich and . Vershik, Random Walks on Discrete Groups: Boundary and Entropy, The Annals of Probability, vol.11, issue.3, pp.457-490, 1983.
DOI : 10.1214/aop/1176993497

A. Vadim, W. Kaimanovich, and . Woess, Boundary and entropy of space homogeneous Markov chains, Ann. Probab, vol.30, issue.1, pp.323-363, 2002.

]. F. Led84 and . Ledrappier, Quelques propriétés des exposants caractéristiques In École d'été de probabilités de Saint-Flour, XII?1982, Lecture Notes in Math, vol.1097, pp.305-396, 1984.

]. F. Led96 and . Ledrappier, Profil d'entropie dans le cas continu, Astérisque, issue.236, pp.189-198, 1996.

[. Ledrappier, Linear Drift and Entropy for Regular Covers, Geometric and Functional Analysis, vol.41, issue.3, pp.710-725, 2010.
DOI : 10.1007/s00039-010-0080-9

URL : https://hal.archives-ouvertes.fr/hal-00559843

T. Lyons and D. Sullivan, Function theory, random paths and covering spaces, Journal of Differential Geometry, vol.19, issue.2, pp.299-323, 1984.
DOI : 10.4310/jdg/1214438681

L. [. Ledrappier and . Shu, Entropy rigidity of symmetric spaces without focal points. ArXiv e-prints, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01019601

P. Lu, Convergence of fundamental solutions of linear parabolic equations under Cheeger???Gromov convergence, Mathematische Annalen, vol.138, issue.1, pp.193-217, 2012.
DOI : 10.1007/s00208-011-0679-7

P. Petersen, Riemannian geometry, volume 171 of Graduate Texts in Mathematics, 2006.

M. S. Pinsker, Information and information stability of random variables and processes. Holden-Day Series in Time Series Analysis, 1964.

J. Prat, Étude asymptotique du mouvement brownien sur une variété riemannienne à courbure négative, C. R. Acad. Sci. Paris Sér. A-B, vol.272, pp.1586-1589, 1971.

J. Prat, Étude asymptotique et convergence angulaire du mouvement brownien sur une variété à courbure négative, C. R. Acad. Sci. Paris Sér. A-B, vol.280, issue.22, pp.1539-1542, 1975.

G. Reeb, Variétés feuilletées, feuilles voisines, C. R. Acad. Sci. Paris, vol.224, pp.1613-1614, 1947.

G. Reeb, Remarque sur les variétés feuilletées contenant une feuille compacte à groupe de Poincaré fini, C. R. Acad. Sci. Paris, vol.226, pp.1337-1339, 1948.

]. V. Roh52 and . Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, issue.71, p.195255, 1952.

T. Sakai, On continuity of injectivity radius function, Math. J. Okayama Univ, vol.25, issue.1, pp.91-97, 1983.

R. M. Solovay, A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable, The Annals of Mathematics, vol.92, issue.1, pp.1-56, 1970.
DOI : 10.2307/1970696

J. and M. Steele, Kingman's subadditive ergodic theorem, Ann. Inst. H. Poincaré Probab. Statist, vol.25, issue.1, pp.93-98, 1989.

N. Th and . Varopoulos, Information theory and harmonic functions, Bull. Sci. Math, vol.110, issue.24, pp.347-389, 1986.

X. Wang, Compactifications of complete Riemannian manifolds and their applications In Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM), vol.20, pp.517-529, 2011.

C. William and . Waterhouse, The absolute-value estimate for symmetric multilinear forms, Linear Algebra Appl, vol.128, pp.97-105, 1990.