Stochastic particle approximations for generalized Boltzmann models and convergence estimates

Graham, C. D.; Méléard, Sylvie

doi:10.1214/aop/1024404281

Cited by 125 publications

(130 citation statements)

References 18 publications

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“…In both cases, we denote by V β,n = (V β,1n , ..., V β,nn ) this Markov process. We use a convergence result proved in Graham-Méléard [6], Theorem 3.1 and obtain a strong approximation result. For a given T > 0, let us denote by |.| T the total variation norm in the space of signed measures on D([0, T ], R 3 ).…”

Section: Theorem 32 Let Us Assume Thatmentioning

confidence: 99%

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

Fournier

2001

Math. Comp.

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Abstract. Using the main ideas of Tanaka, the measure-solution {Pt}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 {Pt}t of solutions {P l t }t of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-Méléard is used and allows us to approximate {P l t }t with the empirical measure {µ 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.

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Section: Theorem 32 Let Us Assume Thatmentioning

confidence: 99%

“…We use a result of Graham-Méléard [6] who discuss this problem in a general context, but in a cutoff case. Thus we consider first cutoff approximations of our model and associate with each cutoff model some cutoff approximating interacting particle systems.…”

Section: Introductionmentioning

confidence: 99%

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

Fournier

2001

Math. Comp.

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“…More recently Graham and Méléard [10] proved propagation of chaos for the more general Povzner equation, which is a kind of mollified spatially heterogeneous Boltzmann equation where each particle can collide with any particle in a small spatial neighbourhood around it. Yet these essentially qualitative methods hardly give quantitative results.…”

Section: Comparison To Older Resultsmentioning

confidence: 99%

“…Yet its proof, whose main ingredient is the use of the utility function U, would work as well in a finite-dimensional setting. So let us imagine that we replaceḢ 9) whereas the exact result is 10) where "≈" means "having equivalent logarithms".…”

Section: Optimalitymentioning

confidence: 99%

Some Ideas About Quantitative Convergence of Collision Models to Their Mean Field Limit

Peyre

2009

J Stat Phys

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We consider a stochastic N -particle model for the spatially homogeneous Boltzmann evolution and prove its convergence to the associated Boltzmann equation when N −→ ∞. For any time T > 0 we bound the distance between the empirical measure of the particle system and the measure given by the Boltzmann evolution in some homogeneous negative Sobolev space. The control we get is Gaussian, i.e. we prove that the distance is bigger than xN −1/2 with a probability of type O(e −x 2 ). The two main ingredients are first a control of fluctuations due to the discrete nature of collisions, secondly a Lipschitz continuity for the Boltzmann collision kernel. The latter condition, in our present setting, is only satisfied for Maxwellian models. Numerical computations tend to show that our results are useful in practice.

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“…[44], [52], [46], [38], [39], [42] and references therein. Some particular combinations of differential and integral generators are mostly centered around spatially nontrivial versions of Boltzmann and Smoluchovski models, see [25], [27], [50], [35] and references therein, see also [23] for models with spontaneous births.…”

Section: Content Of the Paper And Bibliographical Commentsmentioning

confidence: 99%

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

Kolokoltsov

2006

J Stat Phys

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Semigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Euclidean space R d (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulation-fragmentation models.

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Stochastic particle approximations for generalized Boltzmann models and convergence estimates

Cited by 125 publications

References 18 publications

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

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

Some Ideas About Quantitative Convergence of Collision Models to Their Mean Field Limit

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

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