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

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

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


Code(s) de Classification MSC:

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.

Mots Clés: Stable random variables ; diffusion processes ; drifts ; real hyperbolic space ; confluent hypergeometric functions ; Bateman function ; Tricomi function ; Pearson distributions ; perpetuities ; Brownian laws ; Bessel laws

Date: 1999-07-06

Prépublication numéro: PMA-516