# Quantitative concentration inequalities on sample path space for mean field interaction

ESAIM: Probability and Statistics (2010)

- Volume: 14, page 192-209
- ISSN: 1292-8100

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topBolley, François. "Quantitative concentration inequalities on sample path space for mean field interaction." ESAIM: Probability and Statistics 14 (2010): 192-209. <http://eudml.org/doc/250821>.

@article{Bolley2010,

abstract = {
We consider the approximation of a mean field stochastic process by a large interacting particle system. We derive non-asymptotic large deviation bounds
measuring the concentration of the empirical measure of the paths of the particles around the law of the process. The method is based on a coupling argument, strong integrability estimates on the paths in Hölder norm, and a general concentration result for the empirical measure of identically distributed independent paths.
},

author = {Bolley, François},

journal = {ESAIM: Probability and Statistics},

keywords = {Mean field limits; particle approximation; transportation inequalities; mean field limits},

language = {eng},

month = {7},

pages = {192-209},

publisher = {EDP Sciences},

title = {Quantitative concentration inequalities on sample path space for mean field interaction},

url = {http://eudml.org/doc/250821},

volume = {14},

year = {2010},

}

TY - JOUR

AU - Bolley, François

TI - Quantitative concentration inequalities on sample path space for mean field interaction

JO - ESAIM: Probability and Statistics

DA - 2010/7//

PB - EDP Sciences

VL - 14

SP - 192

EP - 209

AB -
We consider the approximation of a mean field stochastic process by a large interacting particle system. We derive non-asymptotic large deviation bounds
measuring the concentration of the empirical measure of the paths of the particles around the law of the process. The method is based on a coupling argument, strong integrability estimates on the paths in Hölder norm, and a general concentration result for the empirical measure of identically distributed independent paths.

LA - eng

KW - Mean field limits; particle approximation; transportation inequalities; mean field limits

UR - http://eudml.org/doc/250821

ER -

## References

top- S. Benachour, B. Roynette, D. Talay and P. Vallois, Nonlinear self-stabilizing processes. I: Existence, invariant probability, propagation of chaos. Stoch. Proc. Appl.75 (1998) 173–201. Zbl0932.60063
- D. Benedetto, E. Caglioti, J.A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys.91 (1998) 979–990. Zbl0921.60057
- S. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999) 1–28. Zbl0924.46027
- F. Bolley, Quantitative concentration inequalities on sample path space for mean field interaction. Available online at www.ceremade.dauphine.fr/~bolley (2008). Zbl1208.82038
- F. Bolley and C. Villani, Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. Ann. Fac. Sci. Toulouse Math.6 (2005) 331–352. Zbl1087.60008
- F. Bolley, A. Guillin and C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory Relat. Fields137 (2007) 541–593. Zbl1113.60093
- J.A. Carrillo, R.J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rat. Mech. Anal.179 (2006) 217–263. Zbl1082.76105
- P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non uniformly case. Probab. Theory Relat. Fields140 (2008) 19–40. Zbl1169.35031
- A. Dembo and O. Zeitouni, Large deviations techniques and applications. Springer, NewYork (1998). Zbl0896.60013
- H. Djellout, A. Guillin and L. Wu, Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab.32 (2004) 2702–2732. Zbl1061.60011
- J. Dolbeault, Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (large time behavior and steady states). J. Math. Pures Appl.9 (1999) 121–157. Zbl1115.82316
- X. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes. Lect. Notes Math. 480. Springer, Berlin (1975). Zbl0331.60025
- N. Gozlan, Principe conditionnel de Gibbs pour des contraintes fines approchées et inégalités de transport. Thèse de doctorat de l'Université de Paris 10-Nanterre, 2005).
- S.R. Kulkarni and O. Zeitouni, A general classification rule for probability measures. Ann. Statist.23 (1995) 1393–1407. Zbl0841.62011
- G.G. Lorentz, Approximation of functions. Holt, Rinehart and Winston, New York (1966). Zbl0153.38901
- F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's. Stoch. Proc. Appl.95 (2001) 109–132. Zbl1059.60084
- S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. Lect. Notes Math. 1627. Springer, Berlin (1996). Zbl0864.60077
- A.-S. Sznitman, Topics in propagation of chaos. Lect. Notes Math. 1464. Springer, Berlin (1991). Zbl0732.60114
- A. van der Vaart and J. Wellner, Weak convergence and empirical processes. Springer, Berlin (1995). Zbl0862.60002
- C. Villani, Topics in optimal transportation, volume 58 of Grad. Stud. Math. A.M.S., Providence (2003). Zbl1106.90001

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