Advertisement

Martingale Problems Associated with the Boltzmann Equation

  • J. Horowitz
  • R. L. Karandikar
Chapter
Part of the Progress in Probability book series (PRPR, volume 18)

Abstract

The Boltzmann equation is a nonlinear integro-partial differential equation that is supposed to describe the distribution of positions and velocities of the molecules in a dilute gas as a function of time. It is assumed that only two molecules at a time can collide.

Keywords

Weak Solution Markov Process BOLTZMANN Equation Local Martingale Martingale Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. BASS, Uniqueness in law for pure jump Markov processes, to appear in PTRF.Google Scholar
  2. 2.
    C. CERCIGNANI, The Boltzmann Equation and its Applications. Springer 1988.Google Scholar
  3. 3.
    P. ECHEVERRIA, A criterion for invariant measures of Markov processes, ZW 61, 1982, 1–16.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    S. ETHIER and T. KURTZ, Markov Processes: Characterization and Convergence. Wiley 1986.Google Scholar
  5. 5.
    T. FUNAKI, The diffusion approximation of the Boltzmann equation of Maxwellian molecules. Publ. RIMS, Kyoto Univ. 19, 1983, 841–886.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    T. FUNAKI, A certain class of diffusion processes associated with nonlinear parabolic equations. ZW 67, 1984, 331–348.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    T. FUNAKI, The diffusion approximation of the spatially homogeneous Boltzmann equation. Duke Math. J. 52, 1985, 1–23.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    P. GÉRARD, Solutions globale du problème de Cauchy pour l’équation de Boltzmann. Sem. Bourbaki 40, 1987–1988, no. 699.Google Scholar
  9. 9.
    F. A. GRÜNBAUM, Propagation of chaos for the Boltzmann equation. Arch. Ratl. Mech. Anal. 42, 1971, 323–345.MATHCrossRefGoogle Scholar
  10. 10.
    J. JACOD, Calcul Stochastiques et Problèmes de Martingales. Springer Lect. Notes Math. no. 714, 1979.Google Scholar
  11. 11.
    M. KAC, Foundations of kinetic theory. Proc. Third Berk. Symp. on Math. Stat. Prob. 3, 1956, 171–197.Google Scholar
  12. 12.
    T. KOMATSU, On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type. Proa. Japan. Acad. Sci. ser. A, 58, 1982, 353–356.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    J.-P. LEPELTIER et B. MARCHAL, Problèmes de martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel. Ann. Inst. H. Poincaré, Nouv. Ser. B, 12, 1976, 43–103.Google Scholar
  14. 14.
    R. LIPTSER and A. SHIRYAEV, Statistics of Stochastic Processes, Springer 1977.Google Scholar
  15. 15.
    H. P. McKEAN, Jr., A Class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. 56, 1966, 1907–1911.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    K. OELSCHLÄGER, A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Prob. 12, 1984, 458–479.MATHCrossRefGoogle Scholar
  17. 17.
    D. STR00CK, Diffusion processes associated with Lévy generators. ZW 32, 1975, 109–244.MathSciNetGoogle Scholar
  18. 18.
    D. STR00CK and S.R.S. VARADHAN, Multidimensional Diffusion Processes. Springer 1979.Google Scholar
  19. 19.
    A.-S. SZNITMAN, Équations de type Boltzmann, spatiàlement homogènes. ZW 66, 1984, 559–592.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    H. TANAKA, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. ZW 46, 1978, 67–105.MATHCrossRefGoogle Scholar
  21. 21.
    H. TANAKA, Some probabilistic problems in the spatially homogeneous Boltzmann equation, in Theory and Application of Random Fields, G. Kallianpur (ed.), Springer Lect. Notes Control Info. Sci. no. 49, 1983.Google Scholar
  22. 22.
    H. TANAKA, Stochastic differential equations corresponding to the spatially homogeneous Boltzmann equation of Maxwellian and non-cutoff type. J. Fac. Sci. Univ. Tokyo Sec. IA, 34, 1987, 351–369.MATHGoogle Scholar
  23. 23.
    C. THOMPSON, Mathematical Statistical Mechanics. Princeton Univ. Press 1972.Google Scholar
  24. 24.
    C. TRUESDELL and R. MUNCASTER, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas. Academic Press 1980.Google Scholar
  25. 25.
    K. UCHIYAMA, Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J. 18, 1988, 245–297.MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • J. Horowitz
    • 1
  • R. L. Karandikar
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Indian Statistical Institute Delhi CenterNew DelhiIndia

Personalised recommendations