Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 30F40 Kleinian groups, See also {20H10}
- 58F17 Geodesic and horocycle flows
- 58F18 Relations with foliations
- 58G32 Diffusion processes and stochastic analysis on manifolds
- 60J60 Diffusion processes, See also {58G32}
- 60F05 Central limit and other weak theorems

**Résumé:** Let $\,\G\, $ be a geometrically finite Kleinian group, relative to the hyperbolic space
$\,\H =\H^{d+1}\, $, and let $\,\d\, $ denote the Hausdorff dimension of its limit set, that we suppose
here strictly larger than $\, d/2\, $. We prove a central limit theorem for the geodesic flow on the
manifold $\,\M := \G\sm\H\, $, with respect to the Patterson-Sullivan measure. The argument uses the
ground-state diffusion and its canonical lift to the frame bundle, for which the existence of a
potential operator is proved.

**Mots Clés:** *geodesic flow ; hyperbolic manifold of infinite volume ; diffusion process ; stable
foliation ; spectral gap ; Patterson-Sullivan measure ; central limit theorem*

**Date:** 2000-07-10

**Prépublication numéro:** *PMA-603*