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:**

- 58F17 Geodesic and horocycle flows
- 58G32 Diffusion processes and stochastic analysis on manifolds
- 60J60 Diffusion processes, See also {58G32}

**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. Denote by
$\,\Phi\, $ the eigenfunction of the hyperbolic Laplacian $\,\D\, $, associated with its
first eigenvalue $\, 2\la_0 = \d (\d -d)\, $. \ Sullivan \mbox{already} considered the associated diffusion
$\, Z_t^\Phi\, $ on $\,\H\, $, whose generator is $\:{1\over 2}\,\D^\Phi := {1\over 2}\,\Phi\1\,
\D\circ\Phi - \la_0\: $. We study the asymptotic behavior of this $\,\Phi $-diffusion, showing that
it exits from $\,\H\, $, almost surely when $\,\d\not= d/2\, $, with as exit law
the normalized
Patterson measure when $\,\d\ge d/2\, $, and some absolutely continuous law when
$\,\d < d/2\, $.
Our method relies on a simple construction of the $\,\Phi $-diffusion.

**Mots Clés:** *diffusion process ; hyperbolic space ; Patterson measure*

**Date:** 1999-09-30

**Prépublication numéro:** *PMA-533*