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Niveau: Supérieur, Doctorat, Bac+8

QUANTITATIVE CONCENTRATION INEQUALITIES FOR EMPIRICAL MEASURES ON NON-COMPACT SPACES FRANC¸OIS BOLLEY, ARNAUD GUILLIN, AND CEDRIC VILLANI Abstract. We establish quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle simulations in a model mean field problem. The tools include coupling arguments, as well as regularity and moment estimates for solutions of certain diffusive partial differential equations. Introduction Large stochastic particle systems constitute a popular way to perform numerical simu- lations in many contexts, either because they are used in some physical model (as in e.g. stellar or granular media) or as an approximation of a continuous model (as in e.g. vortex simulation of the Euler equation: see [21, Chapter 5] for instance). For such systems one may wish to establish concentration estimates showing that the behavior of the system is sharply stabilized as the number N of particles goes to infinity. It is natural to search for these estimates in the setting of large (or moderate) deviations, since one wishes to make sure that the numerical method has a very small probability to give wrong results. From a physical perspective, concentration estimates may be useful to establish the validity of a continuous approximation such as a mean-field limit. When one is interested in the asymptotic behavior of just one, or a few observables (such as the mean position...), there are efficient methods, based for instance on concentration of measure theory.

QUANTITATIVE CONCENTRATION INEQUALITIES FOR EMPIRICAL MEASURES ON NON-COMPACT SPACES FRANC¸OIS BOLLEY, ARNAUD GUILLIN, AND CEDRIC VILLANI Abstract. We establish quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle simulations in a model mean field problem. The tools include coupling arguments, as well as regularity and moment estimates for solutions of certain diffusive partial differential equations. Introduction Large stochastic particle systems constitute a popular way to perform numerical simu- lations in many contexts, either because they are used in some physical model (as in e.g. stellar or granular media) or as an approximation of a continuous model (as in e.g. vortex simulation of the Euler equation: see [21, Chapter 5] for instance). For such systems one may wish to establish concentration estimates showing that the behavior of the system is sharply stabilized as the number N of particles goes to infinity. It is natural to search for these estimates in the setting of large (or moderate) deviations, since one wishes to make sure that the numerical method has a very small probability to give wrong results. From a physical perspective, concentration estimates may be useful to establish the validity of a continuous approximation such as a mean-field limit. When one is interested in the asymptotic behavior of just one, or a few observables (such as the mean position...), there are efficient methods, based for instance on concentration of measure theory.

- approximation such
- measure
- let µ
- main results
- such that∫
- moment estimates
- measures
- t2 inequalities
- associated empirical

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Published by | mijec |

Reads | 12 |

Language | English |

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