Skip to main content
SpringerLink
Log in

Topics in propagation of chaos

  • Conference paper
  • First Online:
Ecole d'Eté de Probabilités de Saint-Flour XIX — 1989

Part of the book series: Lecture Notes in Mathematics ((LNMECOLE,volume 1464))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baxter, J. R.-Chacon, R. V.-Jain, N. C.: Weak limits of stopped diffusions, Trans. Amer. Math. Soc., 293, 2, 767–792, (1986).

    Article  MathSciNet  MATH  Google Scholar 

  2. Billingsley, P.: Convergence of Probability Measures, Wiley, New York (1968).

    MATH  Google Scholar 

  3. Calderoni, P.-Pulvirenti, M.: Propagation chaos for Burger's equation, Ann. Inst. H. Poincaré, série A, N.S. 39, 85–97 (1983).

    MathSciNet  MATH  Google Scholar 

  4. Cercignani, C.: The grad limit for a system of soft spheres, Comm. Pure Appl. Math 26, 4 (1983).

    MathSciNet  MATH  Google Scholar 

  5. Dawson, D. A.: Critical dynamics and fluctuations for a mean field model of cooperative behavior, J. Stat. Phys. 31, 29–85 (1978).

    Article  MathSciNet  Google Scholar 

  6. Dawson, D. A.-Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics 20, 247–308 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  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. Dobrushin, R. L.: Prescribing a system of random variables by conditional distributions, Th. Probab. and its Applic. 3, 469 (1970).

    MATH  Google Scholar 

  9. — Vlasov equations, Funct. Anal. and Appl. 13, 115 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  10. Ferland, R., Giroux, G.: Cutoff Boltzmann equation: convergence of the solution, Adv. Appl. Math. 8, 98–107 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  11. Funaki, T.: The diffusion approximation of the spatially homogeneous Boltzmann equation, Duke Math. J. 52, 1–23, (1985).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Chapter  Google Scholar 

  13. Guo, M.-Papanicolaou, G. C.: Self diffusion of interacting Brownian particles, Taniguchi Symp., Katata 1985, 113–151.

    Google Scholar 

  14. Gutkin, E.: Propagation of chaos and the Hopf-Cole transformation, Adv. Appl. Math. 6, 413–421, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  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. — Some probabilistic aspects of the Boltzmann equation, Acta Physica Austraiaca, suppl. X, Springer, 379–400 (1979).

    Google Scholar 

  17. Karandikar, R. L., Horowitz, J.: Martingale problems associated with the Boltzmann equation, preprint, 1989.

    Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  19. Kurtz, T.: Approximation of population processes CMBS-NSF Reg. Conf. Sci. Appl. Math. Vol. 36, Society for Industrial and Applied Mathematics, Philadelphia (1981).

    Book  Google Scholar 

  20. Kusuoka, S.-Tamura, Y.: Gibbs measures with mean field potentials, J. Fac. Sci. Tokyo Univ., sect. 1A, 31, 1, 223–245 (1984).

    MathSciNet  MATH  Google Scholar 

  21. Kotani, S.-Osada, H.: Propagation of chaos for Burgers' equation, J. Math. Soc. Japan, 37, 275–294 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  22. Lanford, O. E., Time evolution of large classical systems, Lecture Notes in Physics 38, 1–111, Springer, Berlin, (1975).

    MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  24. Liggett, T. M.: Interacting particle systems, Springer, Berlin (1985).

    Book  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  26. — A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. 56, 1907–1911 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  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. — Fluctuations in the kinetic theory of gases, Comm. Pure Appl. Math. 28, 435–455 (1975).

    Article  MathSciNet  Google Scholar 

  29. Marchioro, C.-Pulvirenti, M.: Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys. 84, 483–504 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  30. Murata, H.-Tanaka, H.: An inequality for certain functionals of multidimensional probability distributions, Hiroshima Math. J., 4, 75–81 (1974).

    MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  32. — On the propagation of chaos for diffusion processes with coefficients not of average form, Tokyo Jour. Math. 10 (2), 403–418 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  33. Neveu, J.: Arbres et processus de Galton-Watson, Ann. Inst. Henri Poincaré Nouv. Ser. B, 22, 2,199–208 (1986).

    MathSciNet  MATH  Google Scholar 

  34. Oelschläger K.: A law of large numbers for moderately interacting diffusion processes, Z. Wahrscheinlichkeitstheor. Verw. Gebeite 69, 279–322 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Chapter  Google Scholar 

  36. — Propagation of chaos for the two dimensional Navier-Stokes equation.

    Google Scholar 

  37. Scheutzow, M.: Periodic behavior of the stochastic Brusselator in the mean field limit, Prob. Th. Re. Fields 72, 425–462, (1986).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  39. Sznitman, A. S.: Equations de type Boltzmann spatialement homogènes, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 66, 559–592 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  40. — Nonlinear reflecting diffusion process and the propagation of chaos and fluctuations associated, J. Funct. Anal. 56 (3), 311–336 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  41. — A fluctuation result for nonlinear diffusions, infinite dimensional analysis, S. Albeverio, ed., 145–160, Pitman, Boston (1985).

    Google Scholar 

  42. — A propagation of chaos result for Burgers' equation, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 71, 581–613 (1986).

    MathSciNet  MATH  Google Scholar 

  43. — Propagation of chaos for a system of annihilating Brownian spheres, Comm. Pure Appl. Math. 60, 663–690 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  44. — A trajectorial representation for certain nonlinear equations, Astérisque, 157–158, 363–370 (1988).

    Google Scholar 

  45. — A limiting result for the structure of collisions between many independent diffusions, Probab. Th. Rel Fields 81, 353–381 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  46. Sznitman, A. S.-Varadhan, S.R.S.: A multidimensional process involving local time, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 71, 553–579 (1986).

    MathSciNet  MATH  Google Scholar 

  47. Spohn, H.: The dynamics of systems with many particles, statistical mechanics of local equilibrium states (preprint).

    Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  49. Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 46, 67–105 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  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. — Limit theorems for certain diffusion processes with interaction, Taniguchi Symp. S. A. Katata, 469–488 (1982).

    Google Scholar 

  52. Tanaka, H.-Hitsuda, M.: Central limit theorem for a simple model of interacting particles, Hiroshima Math. J. 11, 415–423 (1981).

    MathSciNet  MATH  Google Scholar 

  53. Uchiyama, K.: On the Boltzmann Grad limit for the Broadwell model of the Boltzmann equation, J. Stat. Phys. 52, 331–355 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  54. — Derivation of the Botlzmann equation from particle dynamics, Hiroshima Math. J. 18, 2 (1988).

    MathSciNet  Google Scholar 

  55. Ueno, T.: A class of Markov processes with bounded nonlinear generators, Japanese J. Math. 38, 19–38 (1968).

    MathSciNet  MATH  Google Scholar 

  56. — A path space and the propagation of chaos for Boltzmann's gas model, Proc. Japan Acad. 6 (47) 529–533 (1971).

    Article  MathSciNet  Google Scholar 

  57. Wild, E.: On the Boltzmann equation in the kinetic theory of gases, Proc. Cambridge Phil. Soc., 47, 602–609 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Authors
  1. Alain-Sol Sznitman
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Paul-Louis Hennequin

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag

About this paper

Cite this paper

Sznitman, AS. (1991). Topics in propagation of chaos. In: Hennequin, PL. (eds) Ecole d'Eté de Probabilités de Saint-Flour XIX — 1989. Lecture Notes in Mathematics, vol 1464. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085169

Download citation

  • DOI: https://doi.org/10.1007/BFb0085169

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53841-7

  • Online ISBN: 978-3-540-46319-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation