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

• 58F17 Geodesic and horocycle flows
• 58G32 Diffusion processes and stochastic analysis on manifolds
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.