Quantitative concentration inequalities on sample path space for mean field interaction
ESAIM: Probability and Statistics, Tome 14 (2010), pp. 192-209.

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.

DOI : 10.1051/ps:2008033
Classification : 82C22, 35K55, 90C08
Mots-clés : mean field limits, particle approximation, transportation inequalities 
@article{PS_2010__14__192_0,
     author = {Bolley, Fran\c{c}ois},
     title = {Quantitative concentration inequalities on sample path space for mean field interaction},
     journal = {ESAIM: Probability and Statistics},
     pages = {192--209},
     publisher = {EDP-Sciences},
     volume = {14},
     year = {2010},
     doi = {10.1051/ps:2008033},
     mrnumber = {2741965},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2008033/}
}
TY  - JOUR
AU  - Bolley, François
TI  - Quantitative concentration inequalities on sample path space for mean field interaction
JO  - ESAIM: Probability and Statistics
PY  - 2010
SP  - 192
EP  - 209
VL  - 14
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps:2008033/
DO  - 10.1051/ps:2008033
LA  - en
ID  - PS_2010__14__192_0
ER  - 
%0 Journal Article
%A Bolley, François
%T Quantitative concentration inequalities on sample path space for mean field interaction
%J ESAIM: Probability and Statistics
%D 2010
%P 192-209
%V 14
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps:2008033/
%R 10.1051/ps:2008033
%G en
%F PS_2010__14__192_0
Bolley, François. Quantitative concentration inequalities on sample path space for mean field interaction. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 192-209. doi : 10.1051/ps:2008033. http://archive.numdam.org/articles/10.1051/ps:2008033/

[1] 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. | Zbl

[2] 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. | Zbl

[3] S. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999) 1-28. | Zbl

[4] F. Bolley, Quantitative concentration inequalities on sample path space for mean field interaction. Available online at www.ceremade.dauphine.fr/~bolley (2008). | Numdam

[5] F. Bolley and C. Villani, Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. Ann. Fac. Sci. Toulouse Math. 6 (2005) 331-352. | Numdam | Zbl

[6] F. Bolley, A. Guillin and C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory Relat. Fields 137 (2007) 541-593. | Zbl

[7] 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. | Zbl

[8] P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non uniformly case. Probab. Theory Relat. Fields 140 (2008) 19-40. | Zbl

[9] A. Dembo and O. Zeitouni, Large deviations techniques and applications. Springer, NewYork (1998). | Zbl

[10] 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. | Zbl

[11] 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. | Zbl

[12] X. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes. Lect. Notes Math. 480. Springer, Berlin (1975). | Zbl

[13] 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).

[14] S.R. Kulkarni and O. Zeitouni, A general classification rule for probability measures. Ann. Statist. 23 (1995) 1393-1407. | Zbl

[15] G.G. Lorentz, Approximation of functions. Holt, Rinehart and Winston, New York (1966). | Zbl

[16] F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's. Stoch. Proc. Appl. 95 (2001) 109-132. | Zbl

[17] S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. Lect. Notes Math. 1627. Springer, Berlin (1996). | Zbl

[18] A.-S. Sznitman, Topics in propagation of chaos. Lect. Notes Math. 1464. Springer, Berlin (1991). | Zbl

[19] A. Van Der Vaart and J. Wellner, Weak convergence and empirical processes. Springer, Berlin (1995). | Zbl

[20] C. Villani, Topics in optimal transportation, volume 58 of Grad. Stud. Math. A.M.S., Providence (2003). | Zbl

Cité par Sources :