no. 6
A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 1078-1089

ℒ denotes a (compact, nonsingular) lamination by hyperbolic Riemann surfaces. We prove that a probability measure on ℒ is harmonic if and only if it is the projection of a measure on the unit tangent bundle T 1 ℒ of ℒ which is invariant under both the geodesic and the horocycle flows.

ℒ denote une lamination (compacte, nonsingulière) par surfaces de Riemann hyperboliques. On montre qu’ une mesure sur ℒ est harmonique si et seulement si elle est la projection d’une mesure sur le fibré tangent unitaire T 1 ℒ qui est invariante sous les flots géodesique et horocyclique.

DOI : 10.1214/07-AIHP147
Classification : 37C12, 58J65, 37D40
Keywords: foliated spaces, harmonic measures, brownian motion on the hyperbolic plane, geodesic flow, horocycle flow 
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Bakhtin, Yuri; Martánez, Matilde. A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 1078-1089. doi: 10.1214/07-AIHP147

[1] C. Bonatti and X. Gómez-Mont. Sur le comportement statistique des feuilles de certains feuilletages holomorphes. Essays on geometry and related topics, Vol. 1, 2. Monogr. Enseign. Math. 38 15-41. Enseignement Math., Geneva, 2001. | Zbl | MR

[2] C. Bonatti, X. Gómez-Mont and R. Vila-Feyer. The foliated geodesic flow on Riccati equations, 2001. Preprint. | MR

[3] A. Candel. Uniformization of surface laminations. Ann. Sci. École Norm. Sup. (4) 26 (1993) 489-516. | Zbl | MR | Numdam

[4] A. Candel. The harmonic measures of Lucy Garnett. Adv. Math. 176 (2003) 187-247. | Zbl | MR

[5] I. Chavel. Eigenvalues in Riemannian Geometry. Academic Press Inc., Orlando, FL, 1984 (including a chapter by Burton Randol, with an appendix by Jozef Dodziuk). | Zbl | MR

[6] B. Deroin and V. Kleptsyn. Random conformal dynamical systems. Geom. Funct. Anal. (2006). To appear. | Zbl | MR

[7] L. Garnett. Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51 (1983) 285-311. | Zbl | MR

[8] M. Heins. Selected Topics in the Classical Theory of Functions of a Complex Variable. Holt, Rinehart and Winston, New York, 1962. | MR

[9] K. Itô and H. P. Mckean, Jr. Diffusion Processes and Their Sample Paths. Springer, Berlin, 1974 (second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125). | Zbl | MR

[10] P. Jiménez. Un subconjunto particular de la variedad de representaciones n-dimensional Rn(Gg). Thesis, Centro de Investigación en Matemáticas, A.C., 2006.

[11] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1988. | Zbl | MR

[12] A. Manning. Dynamics of geodesic and horocycle flows on surfaces of constant negative curvature. Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces (Trieste, 1989) 71-91. Oxford Sci. Publ., Oxford Univ. Press, New York, 1991. | Zbl | MR

[13] P. March. Brownian motion and harmonic functions on rotationally symmetric manifolds. Ann. Probab. 14 (1986) 793-801. | Zbl | MR

[14] M. Martínez. Measures on hyperbolic surface laminations. Ergodic Theory Dynam. Systems 26 (2006) 847-867. | Zbl | MR

[15] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999. | Zbl | MR

[16] H. Thorisson. Coupling, Stationarity, and Regeneration. Springer, New York, 2000. | Zbl | MR

[17] R. J. Zimmer. Ergodic Theory and Semisimple Groups. Birkhäuser, Basel, 1984. | Zbl | MR

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