Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T20:20:53.979Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantitative concentration inequalities onsample path space for mean field interaction

Published online by Cambridge University Press:  29 July 2010

François Bolley
François Bolley*
Affiliation:
Université Paris-Dauphine, Ceremade, UMR 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
*
[email protected]
Get access

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.

Keywords

Mean field limitsparticle approximationtransportation inequalities
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 192 - 209
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benachour, S., Roynette, B., Talay, D. and Vallois, P., Nonlinear self-stabilizing processes. I: Existence, invariant probability, propagation of chaos. Stoch. Proc. Appl. 75 (1998) 173–201. CrossRef
Benedetto, D., Caglioti, E., Carrillo, J.A. and Pulvirenti, M., A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91 (1998) 979–990. CrossRef
S. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999) 1–28.
F. Bolley, Quantitative concentration inequalities on sample path space for mean field interaction. Available online at www.ceremade.dauphine.fr/~bolley (2008).
Bolley, F. and Villani, C., Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. Ann. Fac. Sci. Toulouse Math. 6 (2005) 331–352. CrossRef
Bolley, F., Guillin, A. and Villani, C., Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory Relat. Fields 137 (2007) 541–593. CrossRef
Carrillo, J.A., McCann, R.J. and Villani, C., Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rat. Mech. Anal. 179 (2006) 217–263. CrossRef
Cattiaux, P., Guillin, A. and Malrieu, F., Probabilistic approach for granular media equations in the non uniformly case. Probab. Theory Relat. Fields 140 (2008) 19–40. CrossRef
A. Dembo and O. Zeitouni, Large deviations techniques and applications. Springer, NewYork (1998).
Djellout, H., Guillin, A. and Transportation, L. Wu cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 (2004) 2702–2732.
Dolbeault, J., 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. CrossRef
X. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes. Lect. Notes Math. 480. Springer, Berlin (1975).
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).
Kulkarni, S.R. and Zeitouni, O., A general classification rule for probability measures. Ann. Statist. 23 (1995) 1393–1407. CrossRef
G.G. Lorentz, Approximation of functions. Holt, Rinehart and Winston, New York (1966).
Malrieu, F., Logarithmic Sobolev inequalities for some nonlinear PDE's. Stoch. Proc. Appl. 95 (2001) 109–132. CrossRef
S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. Lect. Notes Math. 1627. Springer, Berlin (1996).
A.-S. Sznitman, Topics in propagation of chaos. Lect. Notes Math. 1464. Springer, Berlin (1991).
A. van der Vaart and J. Wellner, Weak convergence and empirical processes. Springer, Berlin (1995).
C. Villani, Topics in optimal transportation, volume 58 of Grad. Stud. Math. A.M.S., Providence (2003).