Soit M une variété riemannienne complète connexe de courbure sectionnelle bornée. Si M est le revêtement régulier d'une variété de volume fini, alors il n'y a pas de fonctions harmoniques bornées non constantes si, et seulement si, la vitesse de fuite du mouvement brownien est nulle.
Let M be a complete connected Riemannian manifold with bounded sectional curvature. Under the assumption that M is a regular covering of a manifold with finite volume, we establish that M is Liouville if, and only if, the linear rate of escape of Brownian motion on M vanishes.
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@article{CRMATH_2007__344_11_685_0,
author = {Anders Karlsson and Fran\c{c}ois Ledrappier},
title = {Propri\'et\'e de {Liouville} et vitesse de fuite du mouvement brownien},
journal = {Comptes Rendus. Math\'ematique},
pages = {685--690},
publisher = {Elsevier},
volume = {344},
number = {11},
year = {2007},
doi = {10.1016/j.crma.2007.04.019},
language = {fr},
}
Anders Karlsson; François Ledrappier. Propriété de Liouville et vitesse de fuite du mouvement brownien. Comptes Rendus. Mathématique, Volume 344 (2007) no. 11, pp. 685-690. doi : 10.1016/j.crma.2007.04.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.04.019/
[1] On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math., Volume 1 (1989), pp. 201-213
[2] Discretization of positive harmonic functions on Riemannian manifolds and Martin boundary, Luminy, 1992 (Sémin. Congr.), Volume vol. 1, Soc. Math. France, Paris (1996), pp. 77-92
[3] Liouville property for groups and manifolds, Invent. Math., Volume 155 (2004), pp. 55-80
[4] Random walks and discrete subgroups of Lie groups, Adv. Probab. Related Topics, Volume 1 (1971), pp. 1-63
[5] Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.), Volume 36 (1999), pp. 135-249
[6] Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire, Astérisque, Volume 74 (1980), pp. 47-98
[7] Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, vol. 24, 1989
[8] Brownian motion and harmonic functions on covering manifolds. An entropic approach, Dokl. Akad. Nauk SSSR, Volume 288 (1986), pp. 1045-1049 (in Russian); English translation: Soviet Math. Dokl., 33, 1986, pp. 812-816
[9] Boundaries of random walks on polycyclic groups and the law of large numbers for solvable Lie groups, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., Volume 112 (1987) no. 4, pp. 93-95 (in Russian); English translation: Vetsnik Leningrad University: Mathematics., 20, 4, 1987, pp. 49-52
[10] Discretization of bounded harmonic functions on Riemannian manifolds and entropy, Nagoya, 1990, de Gruyter, Berlin (1992), pp. 213-223
[11] A. Karlsson, F. Ledrappier, Drift and Poisson boundaries for random walks, Pure Appl. Math. Quat., à paraître
[12] Counterexamples to Liouville-type theorems, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (6), Volume 34 (1979), pp. 39-43 (in Russian); English translation: Moscow Univ. Math. Bull., 1979, pp. 35-39
[13] Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl. (9), Volume 84 (2005) no. 10, pp. 1295-1361
[14] Function theory, random paths and covering spaces, J. Differential Geom., Volume 19 (1984), pp. 299-323
[15] On non-existence of any 0-invariant positive harmonic function, a counterexample to Stroock's conjecture, Comm. Partial Differential Equations, Volume 20 (1995), pp. 1831-1846
[16] Long range estimates for Markov chains, Bull. Sci. Math., Volume 109 (1985), pp. 225-252
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