Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Non-symmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities

Auteur(s):

Code(s) de Classification MSC:

• 60E07 Infinitely divisible distributions; stable distributions
Résumé: We consider on the hyperbolic half-plane $\Pi=\{z\in\C\:\Im z>0\}$ the drifted Brownian motions that are invariant under the orientation-preserving automorphisms that fix the point at infinity $\infty$ and we study their hitting distributions on the boundary portion $\partial\Pi\setminus\{\infty\}$ and on horocycles through $\infty$, i.e. on the horizontal lines $\{\Im z=a\}$ with $a\>0$. We determine explicitly the the characteristic functions of these hitting distributions and, for $a=0$, also the densities. We prove that they are in the domain of attraction of stable laws, whose parameters are given as explicit functions of the drift coefficients and the starting point.Various connections with previous results in risk theory and representations in terms of Bessel processes are also discussed.