Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### A stochastic particle numerical method for 3D Boltzmann equations without cutoff

Auteur(s):

Code(s) de Classification MSC:

• 60J75 Jump processes
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.