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A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces

Yuri Bakhtin; Matilde Martánez

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 6, page 1078-1089
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Bakhtin, Yuri, and Martánez, Matilde. "A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces." Annales de l'I.H.P. Probabilités et statistiques 44.6 (2008): 1078-1089. <http://eudml.org/doc/78003>.

@article{Bakhtin2008,
abstract = {$\mathcal \{L\}$ denotes a (compact, nonsingular) lamination by hyperbolic Riemann surfaces. We prove that a probability measure on $\mathcal \{L\}$ is harmonic if and only if it is the projection of a measure on the unit tangent bundle $T^\{1\}\mathcal \{L\}$ of $\mathcal \{L\}$ which is invariant under both the geodesic and the horocycle flows.},
author = {Bakhtin, Yuri, Martánez, Matilde},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {foliated spaces; harmonic measures; brownian motion on the hyperbolic plane; geodesic flow; horocycle flow; Brownian motion on the hyperbolic plane},
language = {eng},
number = {6},
pages = {1078-1089},
publisher = {Gauthier-Villars},
title = {A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces},
url = {http://eudml.org/doc/78003},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Bakhtin, Yuri
AU - Martánez, Matilde
TI - A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 6
SP - 1078
EP - 1089
AB - $\mathcal {L}$ denotes a (compact, nonsingular) lamination by hyperbolic Riemann surfaces. We prove that a probability measure on $\mathcal {L}$ is harmonic if and only if it is the projection of a measure on the unit tangent bundle $T^{1}\mathcal {L}$ of $\mathcal {L}$ which is invariant under both the geodesic and the horocycle flows.
LA - eng
KW - foliated spaces; harmonic measures; brownian motion on the hyperbolic plane; geodesic flow; horocycle flow; Brownian motion on the hyperbolic plane
UR - http://eudml.org/doc/78003
ER -

References

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  1. [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. Zbl1010.37025MR1929320
  2. [2] C. Bonatti, X. Gómez-Mont and R. Vila-Feyer. The foliated geodesic flow on Riccati equations, 2001. Preprint. MR1929320
  3. [3] A. Candel. Uniformization of surface laminations. Ann. Sci. École Norm. Sup. (4) 26 (1993) 489–516. Zbl0785.57009MR1235439
  4. [4] A. Candel. The harmonic measures of Lucy Garnett. Adv. Math. 176 (2003) 187–247. Zbl1031.58003MR1982882
  5. [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). Zbl0551.53001MR768584
  6. [6] B. Deroin and V. Kleptsyn. Random conformal dynamical systems. Geom. Funct. Anal. (2006). To appear. Zbl1143.37008MR2373011
  7. [7] L. Garnett. Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51 (1983) 285–311. Zbl0524.58026MR703080
  8. [8] M. Heins. Selected Topics in the Classical Theory of Functions of a Complex Variable. Holt, Rinehart and Winston, New York, 1962. MR162913
  9. [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). Zbl0285.60063MR345224
  10. [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. [11] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1988. Zbl0638.60065MR917065
  12. [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. Zbl0753.58023MR1130173
  13. [13] P. March. Brownian motion and harmonic functions on rotationally symmetric manifolds. Ann. Probab. 14 (1986) 793–801. Zbl0593.60078MR841584
  14. [14] M. Martínez. Measures on hyperbolic surface laminations. Ergodic Theory Dynam. Systems 26 (2006) 847–867. Zbl1107.37027MR2237474
  15. [15] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999. Zbl0917.60006MR1725357
  16. [16] H. Thorisson. Coupling, Stationarity, and Regeneration. Springer, New York, 2000. Zbl0949.60007MR1741181
  17. [17] R. J. Zimmer. Ergodic Theory and Semisimple Groups. Birkhäuser, Basel, 1984. Zbl0571.58015MR776417

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