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:**

- 60J75 Jump processes
- 60H10 Stochastic ordinary differential equations, See Also {
- 60K35 Interacting random processes; statistical mechanics type models; percolation theory, See also {82B43, 82C43}
- 82C40 Kinetic theory of gases

**Résumé:** Using the main ideas of Tanaka \cite{Tanaka:78}, the measure solution
$\{P_t\}_t$ of a $3$-dimensional spatially homogeneous Boltzmann
equation of Maxwellian molecules without cutoff is related to a
Poisson-driven stochastic differential equation. Using this tool, the
convergence to $\{P_t\}_t$ of solutions $\{P^l_t\}_t$ of
approximating Boltzmann equations with cutoff is proved. Then, a
result of Graham-M\'el\'eard, \cite{Graham:96} is used, and allows to
approximate $\{P^l_t\}_t$ with the empirical measure
$\{\mu^{l,n}_t\}_t$ of an easily simulable interacting particle
system. Precise rates of convergence are given. A numerical study lies
at the end of the paper.

**Mots Clés:** *Boltzmann equations without cutoff ; Stochastic differential
equations ; Jump measures ; Interacting particle systems*

**Date:** 2000-02-03

**Prépublication numéro:** *PMA-563*

** Postscript file :** PMA-563.ps

** Compressed (gzip) postscript file :**
PMA-563.ps.gz