The Annals of Probability

Stochastic particle approximations for generalized Boltzmann models and convergence estimates

Carl Graham and Sylvie Méléard

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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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Ann. Probab., Volume 25, Number 1 (1997), 115-132.

First available in Project Euclid: 18 June 2002

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F17: Functional limit theorems; invariance principles 47H15 65C05: Monte Carlo methods 76P05: Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05] 82C40: Kinetic theory of gases 82C80: Numerical methods (Monte Carlo, series resummation, etc.)

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


Graham, Carl; Méléard, Sylvie. Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25 (1997), no. 1, 115--132. doi:10.1214/aop/1024404281.

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