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:** In the present paper, we firstly extend the probabilistic
interpretation
of spatially homogeneous
Boltzmann equations without angular cutoff due firstly to Tanaka and
generalized by Fournier-M\'el\'eard, to some soft
potential cases for a large class of initial data. We relate a measure
solution of the Boltzmann
equation to the solution of a
Poisson-driven stochastic differential equation. Then we consider
renormalized such equations which make
prevail the grazing collisions, and we prove the convergence of the
associated
Boltzmann processes to a process related to the Landau equation
initially introduced by Gu\'erin. The
convergence is pathwise and also implies a convergence at the level of
the partial differential equations. An approximation of
a solution of the Landau equation with soft potential via colliding
stochastic particle systems is derived from this result. We then deduce a Monte-Carlo algorithm of
simulation by a conservative particle method following the asymptotics
of grazing collisions. Numerical results are given.

**Mots Clés:** *Boltzmann equations without cutoff and soft potential ; Landau equation with soft potential ; Nonlinear
stochastic differential equations ; Interacting
particle systems ; Monte-Carlo algorithm
*

**Date:** 2001-11-15

**Prépublication numéro:** *PMA-698*

**Postscript file : **PMA-698.ps