Open Access
January 1997 Stochastic particle approximations for generalized Boltzmann models and convergence estimates
Carl Graham, Sylvie Méléard
Ann. Probab. 25(1): 115-132 (January 1997). DOI: 10.1214/aop/1024404281


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.


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Carl Graham. Sylvie Méléard. "Stochastic particle approximations for generalized Boltzmann models and convergence estimates." Ann. Probab. 25 (1) 115 - 132, January 1997.


Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0873.60076
MathSciNet: MR1428502
Digital Object Identifier: 10.1214/aop/1024404281

Primary: 47H15 , 60F17 , 60K35 , 65C05 , 76P05 , 82C40 , 82C80

Keywords: Boltzmann equation , coupling , Monte Carlo algorithms , nonlinear diffusion with jumps , propagation of chaos , random graphs and tress

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 1 • January 1997
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