# Stochastic approximations of the solution of a full Boltzmann equation with small initial data

ESAIM: Probability and Statistics (2010)

- Volume: 2, page 23-40
- ISSN: 1292-8100

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topMeleard, Sylvie. "Stochastic approximations of the solution of a full Boltzmann equation with small initial data." ESAIM: Probability and Statistics 2 (2010): 23-40. <http://eudml.org/doc/197766>.

@article{Meleard2010,

abstract = {
This paper gives an approximation of the solution of the Boltzmann
equation by stochastic interacting particle systems in a case of
cut-off collision operator and small initial data. In this case,
following the ideas of Mischler and Perthame, we prove the existence
and uniqueness of the solution of this equation and also the existence
and uniqueness of the solution of the associated nonlinear martingale
problem.
Then, we first delocalize the interaction by considering a mollified
Boltzmann equation in which the interaction is averaged on cells of
fixed size which cover the space. In this situation, Graham
and Méléard have obtained an approximation of the mollified
solution by some stochastic interacting particle systems.
Then we consider systems in which the size of the cells depends
on the size of the system. We show that the associated empirical
measures converge in law to a deterministic probability measure
whose density flow is the solution of the full Boltzmann equation.
That suggests an algorithm based on the Poisson interpretation
of the integral term for the simulation of this solution.
},

author = {Meleard, Sylvie},

journal = {ESAIM: Probability and Statistics},

keywords = {Boltzmann equation with small initial data /
interacting particle systems / approximation of the solution. ; Boltzmann equation; existence and uniqueness; nonlinear martingale problem; algorithms; associated mean-field interacting particle systems},

language = {eng},

month = {3},

pages = {23-40},

publisher = {EDP Sciences},

title = {Stochastic approximations of the solution of a full Boltzmann equation with small initial data},

url = {http://eudml.org/doc/197766},

volume = {2},

year = {2010},

}

TY - JOUR

AU - Meleard, Sylvie

TI - Stochastic approximations of the solution of a full Boltzmann equation with small initial data

JO - ESAIM: Probability and Statistics

DA - 2010/3//

PB - EDP Sciences

VL - 2

SP - 23

EP - 40

AB -
This paper gives an approximation of the solution of the Boltzmann
equation by stochastic interacting particle systems in a case of
cut-off collision operator and small initial data. In this case,
following the ideas of Mischler and Perthame, we prove the existence
and uniqueness of the solution of this equation and also the existence
and uniqueness of the solution of the associated nonlinear martingale
problem.
Then, we first delocalize the interaction by considering a mollified
Boltzmann equation in which the interaction is averaged on cells of
fixed size which cover the space. In this situation, Graham
and Méléard have obtained an approximation of the mollified
solution by some stochastic interacting particle systems.
Then we consider systems in which the size of the cells depends
on the size of the system. We show that the associated empirical
measures converge in law to a deterministic probability measure
whose density flow is the solution of the full Boltzmann equation.
That suggests an algorithm based on the Poisson interpretation
of the integral term for the simulation of this solution.

LA - eng

KW - Boltzmann equation with small initial data /
interacting particle systems / approximation of the solution. ; Boltzmann equation; existence and uniqueness; nonlinear martingale problem; algorithms; associated mean-field interacting particle systems

UR - http://eudml.org/doc/197766

ER -

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