Limite de champ moyen de systèmes de particules
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2009-2010), Exposé no. 31, 15 p.

On présente des résultats classiques et récents dans l’étude de la limite de champ moyen de systèmes de particules stochastiques en interaction. Ces derniers résultats visent à couvrir une plus grande variété de modèles et obtenir des estimations précises de la convergence et sont mises en lien avec le comportement en temps grand des systèmes considérés.

Bolley, François 1

1 Ceremade, Umr Cnrs 7534 Université Paris-Dauphine Place du Maréchal de Lattre de Tassigny F-75775 Paris cedex 16
@article{SEDP_2009-2010____A31_0,
     author = {Bolley, Fran\c{c}ois},
     title = {Limite de champ moyen de syst\`emes de particules},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:31},
     pages = {1--15},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2009-2010},
     language = {fr},
     url = {http://www.numdam.org/item/SEDP_2009-2010____A31_0/}
}
TY  - JOUR
AU  - Bolley, François
TI  - Limite de champ moyen de systèmes de particules
JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:31
PY  - 2009-2010
SP  - 1
EP  - 15
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/item/SEDP_2009-2010____A31_0/
LA  - fr
ID  - SEDP_2009-2010____A31_0
ER  - 
%0 Journal Article
%A Bolley, François
%T Limite de champ moyen de systèmes de particules
%J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
%Z talk:31
%D 2009-2010
%P 1-15
%I Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://www.numdam.org/item/SEDP_2009-2010____A31_0/
%G fr
%F SEDP_2009-2010____A31_0
Bolley, François. Limite de champ moyen de systèmes de particules. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2009-2010), Exposé no. 31, 15 p. http://www.numdam.org/item/SEDP_2009-2010____A31_0/

[1] M. Agueh, R. Illner et A. Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic Rel. Models 4, 1 (2011), 1–16. | MR | Zbl

[2] S. Benachour, B. Roynette, D. Talay et P. Vallois. Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos. Stoch. Proc. Appl. 75, 2 (1998), 173–201. | MR | Zbl

[3] D. Benedetto, E. Caglioti, J. A. Carrillo et M. Pulvirenti. A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91, 5-6 (1998), 979–990. | MR | Zbl

[4] F. Bolley. Quantitative concentration inequalities on sample path space for mean field interaction. Esaim Prob. Stat. 14 (2010), 192–209. | EuDML | Numdam | MR | Zbl

[5] F. Bolley, J. A. Cañizo et J. A. Carrillo. Stochastic Mean-Field Limit : Non-Lipschitz Forces and Swarming. A paraître dans Math. Mod. Meth. Appl. Sci. (2011). | MR

[6] F. Bolley, J. A. Cañizo et J. A. Carrillo. Mean-field limit for the stochastic Vicsek model. Prépublication (2011). | MR

[7] F. Bolley, A. Guillin et F. Malrieu. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. Math. Mod. Num. Anal 44, 5 (2010), 867–884. | EuDML | Numdam | MR | Zbl

[8] F. Bolley, A. Guillin et C. Villani. Quantitative concentration inequalities for empirical measures on non-compact spaces. Prob. Theor. Rel. Fields 137, 3-4 (2007), 541–593. | MR | Zbl

[9] W. Braun et K. Hepp. The Vlasov Dynamics and Its Fluctuations in the 1/N Limit of Interacting Classical Particles. Commun. Math. Phys. 56 (1977), 101–113. | MR | Zbl

[10] J. A. Cañizo, J. A. Carrillo et J. Rosado. A well-posedness theory in measures for some kinetic models of collective motion. Math. Mod. Meth. Appl. Sci. 21 (2011), 515–539. | MR | Zbl

[11] E. A. Carlen, M. C. Carvalho, M. Loss, J. Le Roux et C. Villani. Entropy and chaos in the Kac model. Kinetic Rel. Models 3, 1 (2010), 85–122. | MR | Zbl

[12] J. A. Carrillo, M. Fornasier, G. Toscani et F. Vecil. Particle, Kinetic, and Hydrodynamic Models of Swarming. In Naldi, G., Pareschi, L., Toscani, G. (eds.) Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Series : Modelling and Simulation in Science and Technology, Birkhauser, (2010), 297–336. | MR | Zbl

[13] J. A. Carrillo, R. J. McCann et C. Villani. Kinetic equilibration rates for granular media and related equations : entropy dissipation and mass transportation estimates. Rev. Mat. Ibero. 19, 3 (2003), 971–1018. | MR | Zbl

[14] P. Cattiaux, A. Guillin et F. Malrieu. Probabilistic approach for granular media equations in the non uniformly case. Prob. Theor. Rel. Fields 140, 1-2 (2008), 19–40. | MR | Zbl

[15] F. Cucker et S. Smale. Emergent behavior in flocks. IEEE Trans. Automat. Control 52 (2007), 852–862. | MR

[16] R. Dobrushin. Vlasov equations. Funct. Anal. Appl.13 (1979), 115–123. | MR | Zbl

[17] M. R. D’Orsogna, Y. L. Chuang, A. L. Bertozzi et L. Chayes. Self-propelled particles with soft-core interactions : patterns, stability, and collapse. Phys. Rev. Lett. 96, 2006.

[18] F. Golse The mean-field limit for the dynamics of large particle systems, Journées équations aux dérivées partielles, Forges-les-Eaux (2003), 1–47. | Numdam | MR | Zbl

[19] F. Golse. The mean-field limit for a regularized Vlasov-Maxwell dynamics. Prépublication (2010).

[20] M. Hauray et P.-E. Jabin. N particles approximation of the Vlasov equations with singular potential. Arch. Rach. Mech. Anal. 183, 3 (2007), 489–524. | MR | Zbl

[21] G. Loeper. Uniqueness of the solution to the Vlasov-Poisson system with bounded density. J. Math. Pures Appl. 9, 86 (2006), 68–79. | MR | Zbl

[22] F. Malrieu. Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stoch. Proc. Appl. 95, 1 (2001), 109–132. | MR | Zbl

[23] F. Malrieu. Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13, 2 (2003), 540–560. | MR | Zbl

[24] H. P. McKean. Propagation of chaos for a class of non-linear parabolic equations. In Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967. | Zbl

[25] S. Méléard. Asymptotic behaviour of some interacting particle systems ; McKean-Vlasov and Boltzmann models. Lecture Notes in Math. 1627, Springer, Berlin, 1996. | MR | Zbl

[26] H. Neunzert. An introduction to the nonlinear Boltzmann-Vlasov equation. Lecture Notes in Math. 1048. Springer, Berlin, 1984. | MR | Zbl

[27] A.-S. Sznitman. Topics in propagation of chaos. Lecture Notes in Math. 1464, Springer, Berlin, 1991. | MR | Zbl

[28] D. Talay. Probabilistic numerical methods for partial differential equations : elements of analysis. Lecture Notes in Math. 1627, Springer, Berlin, 1996. | MR | Zbl

[29] T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen et O. Shochet. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, (1995), 1226–1229.

[30] C. Villani. Optimal transport, old and new. Grundlehren der math. Wiss. 338, Springer, Berlin, 2009. | MR | Zbl

[31] C. Yates, R. Erban, C. Escudero, L. Couzin, J. Buhl, L. Kevrekidis, P. Maini et D. Sumpter. Inherent noise can facilitate coherence in collective swarm motion. Proc. Nat. Acad. Sci. 106, 14 (2009), 5464–5469.