Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces

Yuri Bakhtin and Matilde Martínez

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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.

Résumé

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

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 44, Number 6 (2008), 1078-1089.

Dates
First available in Project Euclid: 21 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1227287565

Digital Object Identifier
doi:10.1214/07-AIHP147

Mathematical Reviews number (MathSciNet)
MR2469335

Zentralblatt MATH identifier
1189.37033

Subjects
Primary: 37C12 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Keywords
Foliated spaces harmonic measures Brownian Motion on the hyperbolic plane geodesic flow horocycle flow

Citation

Bakhtin, Yuri; Martínez, Matilde. A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 6, 1078--1089. doi:10.1214/07-AIHP147. https://projecteuclid.org/euclid.aihp/1227287565


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