Topics in propagation of chaos

  • Alain-Sol Sznitman
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1464)


Brownian Motion Hard Sphere Nonlinear Process Poisson Point Process Interact Particle System 
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  1. 1.
    Baxter, J. R.-Chacon, R. V.-Jain, N. C.: Weak limits of stopped diffusions, Trans. Amer. Math. Soc., 293, 2, 767–792, (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Billingsley, P.: Convergence of Probability Measures, Wiley, New York (1968).zbMATHGoogle Scholar
  3. 3.
    Calderoni, P.-Pulvirenti, M.: Propagation chaos for Burger's equation, Ann. Inst. H. Poincaré, série A, N.S. 39, 85–97 (1983).MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cercignani, C.: The grad limit for a system of soft spheres, Comm. Pure Appl. Math 26, 4 (1983).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dawson, D. A.: Critical dynamics and fluctuations for a mean field model of cooperative behavior, J. Stat. Phys. 31, 29–85 (1978).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dawson, D. A.-Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics 20, 247–308 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Masi, A.-Ianiro, N.-Pellegrinotti, A.-Pressutti, E.: A survey of the hydrodynamical behavior of many particle systems, in Nonequilibrium phenomena II, Ed.: J. L. Lebowitz, E. W. Montroll, Elsevier (1984).Google Scholar
  8. 8.
    Dobrushin, R. L.: Prescribing a system of random variables by conditional distributions, Th. Probab. and its Applic. 3, 469 (1970).zbMATHGoogle Scholar
  9. 9.
    — Vlasov equations, Funct. Anal. and Appl. 13, 115 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ferland, R., Giroux, G.: Cutoff Boltzmann equation: convergence of the solution, Adv. Appl. Math. 8, 98–107 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Funaki, T.: The diffusion approximation of the spatially homogeneous Boltzmann equation, Duke Math. J. 52, 1–23, (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goodman, J., Convergence of the random value method, IMA, vol. 9, G. Papanciolaou ed., Hydrodynamic Behavior and Interacting Particles, 99–106, Springer, Berlin (1987).CrossRefGoogle Scholar
  13. 13.
    Guo, M.-Papanicolaou, G. C.: Self diffusion of interacting Brownian particles, Taniguchi Symp., Katata 1985, 113–151.Google Scholar
  14. 14.
    Gutkin, E.: Propagation of chaos and the Hopf-Cole transformation, Adv. Appl. Math. 6, 413–421, 1985.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kac, M.: Foundation of kinetic theory, Proc. Third Berkeley Symp. on Math. Stat. and Probab. 3, 171–197, Univ. of Calif. Press (1956).Google Scholar
  16. 16.
    — Some probabilistic aspects of the Boltzmann equation, Acta Physica Austraiaca, suppl. X, Springer, 379–400 (1979).Google Scholar
  17. 17.
    Karandikar, R. L., Horowitz, J.: Martingale problems associated with the Boltzmann equation, preprint, 1989.Google Scholar
  18. 18.
    Kipnis, C.-Olla, S.-Varadhan, S.R.S.: Hydrodynamics and large deviation for simple exclusion processes Comm. Pure Appl. Math., 42, 115–137, (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kurtz, T.: Approximation of population processes CMBS-NSF Reg. Conf. Sci. Appl. Math. Vol. 36, Society for Industrial and Applied Mathematics, Philadelphia (1981).CrossRefGoogle Scholar
  20. 20.
    Kusuoka, S.-Tamura, Y.: Gibbs measures with mean field potentials, J. Fac. Sci. Tokyo Univ., sect. 1A, 31, 1, 223–245 (1984).MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kotani, S.-Osada, H.: Propagation of chaos for Burgers' equation, J. Math. Soc. Japan, 37, 275–294 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lanford, O. E., Time evolution of large classical systems, Lecture Notes in Physics 38, 1–111, Springer, Berlin, (1975).zbMATHGoogle Scholar
  23. 23.
    Lang, R.-Nguyen, X.X.: Smoluchowski's theory of coagulation in colloids holds rigorously in the Boltzmann-Grad limit, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 54, 227–280 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liggett, T. M.: Interacting particle systems, Springer, Berlin (1985).CrossRefzbMATHGoogle Scholar
  25. 25.
    McKean, H. P.: Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas, Arch. Rational Mech. Anal. 21, 347–367 (1966).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    — A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. 56, 1907–1911 (1966).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    — Propagation of chaos for a class of nonlinear parabolic equations, Lecture series in differential equations 7, 41–57, Catholic University, Washington, D. C. (1967).Google Scholar
  28. 28.
    — Fluctuations in the kinetic theory of gases, Comm. Pure Appl. Math. 28, 435–455 (1975).MathSciNetCrossRefGoogle Scholar
  29. 29.
    Marchioro, C.-Pulvirenti, M.: Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys. 84, 483–504 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Murata, H.-Tanaka, H.: An inequality for certain functionals of multidimensional probability distributions, Hiroshima Math. J., 4, 75–81 (1974).MathSciNetzbMATHGoogle Scholar
  31. 31.
    Nagasawa, M.-Tanaka, H.: Diffusion with interaction and collisions between colored particles and the propagation of chaos, Probab. Th. Rel. Fields 74, 161–198 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    — On the propagation of chaos for diffusion processes with coefficients not of average form, Tokyo Jour. Math. 10 (2), 403–418 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Neveu, J.: Arbres et processus de Galton-Watson, Ann. Inst. Henri Poincaré Nouv. Ser. B, 22, 2,199–208 (1986).MathSciNetzbMATHGoogle Scholar
  34. 34.
    Oelschläger K.: A law of large numbers for moderately interacting diffusion processes, Z. Wahrscheinlichkeitstheor. Verw. Gebeite 69, 279–322 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Osada, H.: Limit points of empirical distributions of vortices with small viscosity, IMA, vol. 9, G. Papanicolaou ed., Hydrodynamic behavior and interacting particles, 117–126, Springer, Berlin (1987).CrossRefGoogle Scholar
  36. 36.
    — Propagation of chaos for the two dimensional Navier-Stokes equation.Google Scholar
  37. 37.
    Scheutzow, M.: Periodic behavior of the stochastic Brusselator in the mean field limit, Prob. Th. Re. Fields 72, 425–462, (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Shiga, T.-Tanaka, H.: Central limit theorem for a system of Markovian particles with mean field interactions, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 69, 439–445 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sznitman, A. S.: Equations de type Boltzmann spatialement homogènes, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 66, 559–592 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    — Nonlinear reflecting diffusion process and the propagation of chaos and fluctuations associated, J. Funct. Anal. 56 (3), 311–336 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    — A fluctuation result for nonlinear diffusions, infinite dimensional analysis, S. Albeverio, ed., 145–160, Pitman, Boston (1985).Google Scholar
  42. 42.
    — A propagation of chaos result for Burgers' equation, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 71, 581–613 (1986).MathSciNetzbMATHGoogle Scholar
  43. 43.
    — Propagation of chaos for a system of annihilating Brownian spheres, Comm. Pure Appl. Math. 60, 663–690 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    — A trajectorial representation for certain nonlinear equations, Astérisque, 157–158, 363–370 (1988).Google Scholar
  45. 45.
    — A limiting result for the structure of collisions between many independent diffusions, Probab. Th. Rel Fields 81, 353–381 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Sznitman, A. S.-Varadhan, S.R.S.: A multidimensional process involving local time, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 71, 553–579 (1986).MathSciNetzbMATHGoogle Scholar
  47. 47.
    Spohn, H.: The dynamics of systems with many particles, statistical mechanics of local equilibrium states (preprint).Google Scholar
  48. 48.
    Tamura, Y.: On asymptotic behaviors of the solution of a nonlinear diffusion equation, J. Fac. Sci. Tokyo Univ., sect. IA, 31, 1, 195–221 (1984).MathSciNetzbMATHGoogle Scholar
  49. 49.
    Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 46, 67–105 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    — Some probabilistic problems in the spatially homogeneous Boltzmann equation, Proc. IFIP-ISI conf. on appl. of random fields, Bangalore, Jan. 82.Google Scholar
  51. 51.
    — Limit theorems for certain diffusion processes with interaction, Taniguchi Symp. S. A. Katata, 469–488 (1982).Google Scholar
  52. 52.
    Tanaka, H.-Hitsuda, M.: Central limit theorem for a simple model of interacting particles, Hiroshima Math. J. 11, 415–423 (1981).MathSciNetzbMATHGoogle Scholar
  53. 53.
    Uchiyama, K.: On the Boltzmann Grad limit for the Broadwell model of the Boltzmann equation, J. Stat. Phys. 52, 331–355 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    — Derivation of the Botlzmann equation from particle dynamics, Hiroshima Math. J. 18, 2 (1988).MathSciNetGoogle Scholar
  55. 55.
    Ueno, T.: A class of Markov processes with bounded nonlinear generators, Japanese J. Math. 38, 19–38 (1968).MathSciNetzbMATHGoogle Scholar
  56. 56.
    — A path space and the propagation of chaos for Boltzmann's gas model, Proc. Japan Acad. 6 (47) 529–533 (1971).MathSciNetCrossRefGoogle Scholar
  57. 57.
    Wild, E.: On the Boltzmann equation in the kinetic theory of gases, Proc. Cambridge Phil. Soc., 47, 602–609 (1951).MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Zvonkin, A. K.: A transformation of the phase space of a diffusion process that removes the drift, Math. USSR Sbornik, 22, 1, 129–149, (1974).MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag 1991

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  • Alain-Sol Sznitman

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