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:**

- 60E07 Infinitely divisible distributions; stable distributions
- 60J60 Diffusion processes, See also {58G32}
- 33C15 Confluent hypergeometric functions, Whittaker functions,
- 51M10 Hyperbolic and elliptic geometries (general) and generalizations
- 90A99 None of the above but in this section

**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*