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:**

- 60J30 Processes with independent increments
- 60K35 Interacting random processes; statistical mechanics type models; percolation theory, See also {82B43, 82C43}
- 82C21 Dynamic continuum models (systems of particles, etc.)

**Résumé:** We consider the following elementary model for gravitational
clustering
in an expanding universe. At the initial time, there is a doubly
infinite sequence of particles
lying in a one-dimensional universe that is expanding at constant rate.
We suppose that each particle
$p$ attracts points at a certain rate $c(p)/2$ depending only on $p$,
and when two particles, say
$p$ and $q$, collide by the effect of attraction, they merge as a single
particle $p*q$.
The main purpose of this work is to point at the following remarkable
property of Poisson clouds in these
dynamics. Under certain technical conditions, if at the initial time the
system is
distributed according to a spatially stationary Poisson cloud with
intensity $\mu_0$, then at any time
$t>0$, the system will again have a Poissonian distribution, now with
intensity $\mu_t$, where the family
$(\mu_t, t\geq0)$ solves a generalization of Smoluchowski's coagulation
equation.

**Mots Clés:** *Aggregation ; Poisson cloud ; expanding universe ; Smoluchowski's coagulation equation*

**Date:** 2002-02-07

**Prépublication numéro:** *PMA-708*