We specify the Markov process corresponding to a generalized
mollified Boltzmann equation with general motion between collisions and
nonlinear bounded jump (collision) operator, and give the nonlinear martingale
problem it solves. We consider various linear interacting particle systems in
order to approximate this nonlinear process. We prove propagation of chaos, in
variation norm on path space with a precise rate of convergence, using coupling
and interaction graph techniques and a representation of the nonlinear process
on a Boltzmann tree. No regularity nor uniqueness assumption is needed. We then
consider a nonlinear equation with both Vlasov and Boltzmann terms and give a
weak pathwise propagation of chaos result using a compactness-uniqueness method
which necessitates some regularity. These results imply functional laws of
large numbers and extend to multitype models. We give algorithms simulating or
approximating the particle systems.
Publié le : 1997-01-14
Classification:
Boltzmann equation,
nonlinear diffusion with jumps,
random graphs and tress,
coupling,
propagation of chaos,
Monte Carlo algorithms,
60K35,
60F17,
47H15,
65C05,
76P05,
82C40,
82C80
@article{1024404281,
author = {Graham, Carl and M\'el\'eard, Sylvie},
title = {Stochastic particle approximations for generalized Boltzmann
models and convergence estimates},
journal = {Ann. Probab.},
volume = {25},
number = {4},
year = {1997},
pages = { 115-132},
language = {en},
url = {http://dml.mathdoc.fr/item/1024404281}
}
Graham, Carl; Méléard, Sylvie. Stochastic particle approximations for generalized Boltzmann
models and convergence estimates. Ann. Probab., Tome 25 (1997) no. 4, pp. 115-132. http://gdmltest.u-ga.fr/item/1024404281/