On Kac's chaos and related problems

Journal of Functional Analysis

Volumn 266, Issue 10, 2014, Pages 6055-6157

  Hauray, Maxime ^a   Mischler, Stéphane ^b  

^a AIX MARSEILLE UNIVERSITÉ   (France)

^b UNIVERSITÉ PARIS DAUPHINE   (France)

Author keywords

CLT with optimal rate; De Finetti, Hewitt and Savage theorem; Entropy chaos; Fisher information chaos; Kac's chaos; Mean field limit; Monge Kantorovich Wasserstein distance; Probability measures mixtures; Qualitative chaos; Quantitative chaos

Indexed keywords

EID: 84895932069     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2014.02.030     Document Type: Article

References (76)

1. Ajtai M., Komlós J., Tusnády G. On optimal matchings. Combinatorica 1984, 4(4):259-264.
2. Arkeryd L., Caprino S., Ianiro N. The homogeneous Boltzmann hierarchy and statistical solutions to the homogeneous Boltzmann equation. J. Stat. Phys. 1991, 63(1-2):345-361.
3. Artstein S., Ball K.M., Barthe F., Naor A. On the rate of convergence in the entropic central limit theorem. Probab. Theory Related Fields 2004, 129(3):381-390.
4. Barthe F., Bordenave C. Combinatorial optimization over two random point sets arxiv:1103.2734v2.
5. Barthe F., Cordero-Erausquin D., Maurey B. Entropy of spherical marginals and related inequalities. J. Math. Pures Appl. (9) 2006, 86(2):89-99.
6. Ben Arous G., Zeitouni O. Increasing propagation of chaos for mean field models. Ann. Inst. H. Poincaré Probab. Statist. 1999, 35(1):85-102.
7. Berry A.C. The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 1941, 49:122-136.
8. Bobkov S.G., Chistyakow G., Götze F. Berry-Esseen bounds in the entropic central limit theorem arxiv:1104.3994v1.
9. Bobkov S.G., Gentil I., Ledoux M. Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 2001, 80(7):669-696.
10. Boissard E., Le Gouic T. On the mean speed of convergence of empirical and occupation measures in Wasserstein distance arxiv:1105.5263v1.
11. Bolley F., Guillin A., Malrieu F. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. M2AN Math. Model. Numer. Anal. 2010, 44(5):867-884.
12. Bolley F., Guillin A., Villani C. Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory Related Fields 2007, 137(3-4):541-593.
13. Caglioti E., Lions P.-L., Marchioro C., Pulvirenti M. A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Comm. Math. Phys. 1992, 143(3):501-525.
14. Caglioti E., Lions P.-L., Marchioro C., Pulvirenti M. A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. II. Comm. Math. Phys. 1995, 174(2):229-260.
15. Cantelli F.P. Sulla determinazione empirica delle leggi di probabilita. Giorn. Ist. Ital. Attuari 1933, 4:421-424.
16. Carlen E.A. Superadditivity of Fisher's information and logarithmic Sobolev inequalities. J. Funct. Anal. 1991, 101(1):194-211.
17. Carlen E.A., Carvalho M.C., Le Roux J., Loss M., Villani C. Entropy and chaos in the Kac model. Kinet. Relat. Models 2010, 3(1):85-122.
18. N and related entropy inequalities. J. Geom. Anal. 2004, 14(3):487-520.
19. Carrapatoso K. Quantitative and qualitative Kac's chaos on the Boltzmann's sphere arxiv:1205.1241.
20. Carrapatoso K. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules arxiv:1212.3724.
21. Carrillo J., Toscani G. Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Parma 2007, 6:75-198.
22. Choquet G., Meyer P.-A. Existence et unicité des représentations intégrales dans les convexes compacts quelconques. Ann. Inst. Fourier (Grenoble) 1963, 13:139-154.
23. Cordero-Erausquin D. Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 2002, 161(3):257-269.
24. de Finetti B. La prévision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincaré 1937, 7(1):1-68.
25. Diaconis P., Freedman D. A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Probab. Statist. 1987, 23(2 Suppl.):397-423.
26. Dobrić V., Yukich J.E. Asymptotics for transportation cost in high dimensions. J. Theoret. Probab. 1995, 8(1):97-118.
27. Dobrušin R.L. Vlasov equations. Funktsional. Anal. i Prilozhen. 1979, 13(2):48-58. 96.
28. Dudley R.M. The speed of mean Glivenko-Cantelli convergence. Ann. Math. Statist. 1968, 40:40-50.
29. Esseen C.-G. On the Liapounoff limit of error in the theory of probability. Ark. Mat. Astr. Fys. 1942, 28A(9). 19 pp.
30. Feller W. An Introduction to Probability Theory and Its Applications, vol. II 1971, John Wiley & Sons Inc., New York. second edition.
31. Fonseca I., Parry G. Equilibrium configurations of defective crystals. Arch. Ration. Mech. Anal. 1992, 120(3):245-283.
32. N. Fournier, M. Hauray, S. Mischler, Propagation of chaos for the 2d viscous vortex model, hal-00762286.
33. Glivenko V. Sulla determinazione empirica della legge di probabilita. Giorn. Ist. Ital. Attuari 1933, 4:92-99.
34. Goudon T., Junca S., Toscani G. Fourier-based distances and Berry-Esseen like inequalities for smooth densities. Monatsh. Math. 2002, 135(2):115-136.
35. Graham C., Méléard S. Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 1997, 25:115-132.
36. Grünbaum F.A. Propagation of chaos for the Boltzmann equation. Arch. Ration. Mech. Anal. 1971, 42:323-345.
37. Guo M.Z., Papanicolaou G.C., Varadhan S.R.S. Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 1988, 118(1):31-59.
38. Hauray M., Jabin P.-E. N-particles approximation of the Vlasov equations with singular potential. Arch. Ration. Mech. Anal. 2007, 183(3):489-524.
39. M. Hauray, P.-E. Jabin, Particles approximations of Vlasov equations with singular forces, Part 2, hal-00609453.
40. Hewitt E., Savage L.J. Symmetric measures on Cartesian products. Trans. Amer. Math. Soc. 1955, 80:470-501.
41. Kac M. Foundations of kinetic theory. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. III 1956, 171-197. University of California Press.
42. Kosygina E. The behavior of the specific entropy in the hydrodynamic scaling limit. Ann. Probab. 2001, 29(3):1086-1110.
43. Lanford O.E. Time evolution of large classical systems. Lecture Notes in Phys. 1975, vol. 38:1-111. Springer, Berlin.
44. Lions P.-L. Théorie des jeux de champ moyen et applications (mean field games) in: Cours du Collège de France, 2007-2009. http://www.college-de-france.fr/default/EN/all/equ_der/audio_video.jsp.
45. Lions P.-L., Toscani G. A strengthened central limit theorem for smooth densities. J. Funct. Anal. 1995, 129(1):148-167.
46. Lott J., Villani C. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 2009, 169(3):903-991.
47. Malrieu F., Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl. 2001, 95(1):109-132.
48. McKean H.P. Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arch. Ration. Mech. Anal. 1966, 21:343-367.
49. McKean H.P. An exponential formula for solving Boltzmann's equation for a Maxwellian gas. J. Combin. Theory 1967, 2:358-382.
50. McKean H.P. The central limit theorem for Carleman's equation. Israel J. Math. 1975, 21(1):54-92.
51. Mehler F.G. Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaschen Functionen höherer Ordnungn. Crelle's J. 1866, 66:161-176.
52. Méléard S. Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. Lecture Notes in Math. 1996, vol. 1627:42-95. Springer, Berlin.
53. Messer J., Spohn H. Statistical mechanics of the isothermal Lane-Emden equation. J. Stat. Phys. 1982, 29(3):561-578.
54. Mischler S. Le programme de Kac sur les limites de champ moyen. Séminaire EDP-X Décembre 2010.
55. Mischler S., Mouhot C. Kac's program in kinetic theory. Invent. Math. 2014, 193(1):1-147. in press.
56. S. Mischler, C. Mouhot, On chaos properties of a Boltzmann-Kac system for inelastic collisions, in preparation.
57. S. Mischler, C. Mouhot, B. Wennberg, A new approach to quantitative chaos propagation for drift, diffusion and jump processes, hal-00559132.
58. Olla S., Varadhan S.R.S., Yau H.-T. Hydrodynamical limit for a Hamiltonian system with weak noise. Comm. Math. Phys. 1993, 155(3):523-560.
59. Osada H. Propagation of chaos for the two-dimensional Navier-Stokes equation. Proc. Japan Acad. Ser. A Math. Sci. 1986, 62(1):8-11.
60. Otto F. The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 2001, 26(1-2):101-174.
61. Otto F., Villani C. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 2000, 173(2):361-400.
62. Rachev S.T., Rüschendorf L. Mass transportation problems. vol. II. Probab. Appl. (N. Y.) 1998, Springer-Verlag, New York.
63. Rio E. Upper bounds for minimal distances in the central limit theorem. Ann. Inst. H. Poincaré Probab. Statist. 2009, 45(3):802-817.
64. Robinson D.W., Ruelle D. Mean entropy of states in classical statistical mechanics. Comm. Math. Phys. 1967, 5:288-300.
65. Sznitman A.-S. Équations de type de Boltzmann, spatialement homogènes. Z. Wahrsch. Verw. Gebiete 1984, 66(4):559-592.
66. Sznitman A.-S. A propagation of chaos result for Burgers' equation. IMA Vol. Math. Appl. 1987, vol. 9:181-188. Springer, New York.
67. Sznitman A.-S. Topics in propagation of chaos. Lecture Notes in Math. 1991, vol. 1464:165-251. Springer, Berlin.
68. Talagrand M. Matching random samples in many dimensions. Ann. Appl. Probab. 1992, 2(4):846-856.
69. Tanaka H. Some probabilistic problems in the spatially homogeneous Boltzmann equation. Lecture Notes in Control and Inform. Sci. 1983, vol. 49:258-267. Springer, Berlin.
70. Toscani G. New a priori estimates for the spatially homogeneous Boltzmann equation. Contin. Mech. Thermodyn. 1992, 4(2):81-93.
71. Varadarajan V.S. On the convergence of sample probability distributions. Sankhyā 1958, 19:23-26.
72. Villani C. Fisher information estimates for Boltzmann's collision operator. J. Math. Pures Appl. 1998, 77(8):821-837.
73. Villani C. Decrease of the Fisher information for solutions of the spatially homogeneous Landau equation with Maxwellian molecules. Math. Models Methods Appl. Sci. 2000, 10(2):153-161.
74. Villani C. Topics in Optimal Transportation. Grad. Stud. Math. 2003, vol. 58. American Mathematical Society.
75. Villani C. Optimal Transport, Old and New. Grundlehren Math. Wiss. 2009, vol. 338. Springer-Verlag, Berlin.
76. Yau H.-T. Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 1991, 22(1):63-80.