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

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

**Code(s) de Classification MSC:**

- 60H07 Stochastic calculus of variations and the Malliavin calculus
- 60J60 Diffusion processes, See also {58G32}
- 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
- 58F25 Flows

**Résumé:** We consider the invariant measure for finite systems of branching diffusions
with immigration. In case of an absolute continuous immigration measure, we
use the properties of the underlying stochastic flow governing the motion of
every particle in order to show smoothness of the invariant density of the
particle process. In case of a singular immigration measure, the main tool is
Malliavin calculus which allows to show that the density is $C^{\infty}$ in any point which is not an atom of the immigration measure.

**Mots Clés:** *branching diffusions ; invariant measure ; stochastic flows ; Malliavin calculus*

**Date:** 2001-09-27

**Prépublication numéro:** *PMA-688*