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

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Exit law of the fundamental diffusion associated with a Kleinian group

Auteur(s):

Code(s) de Classification MSC:

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