Large-Scale Structure of the Universe and Asymptotics of Burgers' Turbulence with Heavy-tailed Dependent Data Yiming Hue and

  • W. A. Woyczynskil
  • Yiming Hu
Part I: Lectures
Part of the Lecture Notes in Physics book series (LNP, volume 457)


Large-time asymptotics of the statistical solutionsν(t,χ) of the Burgers' equation ν t + ννχ = ννχχ is considered. The initial velocity potential is assumed to be of the shot noise type with dependent amplitudes and heavy tails. The problem arises naturally in the adhesion model of the large-scale distribution of matter in the Universe. As t → ∞, the random field ν(t, χ) becomes stochastically relatively asymptotically equivalent to a field with “saw-tooth” trajectories. The intermittent shocks of the velocity field correspond then to the regions of high density in the coupled passive tracer density field. This paper extends a result of S.Albeverio, S.A.Molchanov and D. Surgailis (1992), obtained for the case of independent amplitudes.


Random Field Shot Noise Burger Equation Heavy Tail Dependent Amplitude 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Albeverio, S.A. Molchanov and D. Surgailis, Stratified structure of the Universe and Burgers` equation-a probabilistic approach, Prob. Theory Rel. Fields 100, (1994) to appear.Google Scholar
  2. 2.
    T. Funaki, D. Surgailis and W.A. Woyczynski, Gibbs-Cox random fields and Burgers` turbulence, Ann. Appl. Probability, 5, 701–735 (1995).Google Scholar
  3. 3.
    S.N. Gurbatov, A.I. Saichev, D.F. Shandarin, Soviet Phys. Dokl., 30, 921 (1985).Google Scholar
  4. 4.
    Y. Hu and W.A. Woyczynski, An extremal rearrangement property of statistical solutions of Burgers` equation, Annals of Applied Probability, 4, No.3, 838–858 (1994).Google Scholar
  5. 5.
    Y. Hu and W.A. Woyczynski, Limit behavior of quadratic forms of moving averages of i.i.d. random variables and statistical solutions of the Burgers` equation. J. Multivariate Anal., 52, 15–44 (1995).Google Scholar
  6. 6.
    Y. Hu and W.A. Woyczynski, Shock density in Burgers' turbulence, in Nonlinear Stochastic PDE's: Burgers' Turbulence and Hydrodynamic Limit, IMA Volumes, Springer-Verlag, pp. 211–226 (1995).Google Scholar
  7. 7.
    L. Kofman, D. Pogosyan, S.F. Shandarin and A.L. Mellot, Coherent structures in the Universe and the adhesion model, Astrophys. J., 393, 437–449 (1992).Google Scholar
  8. 8.
    S. Kwapien and W.A. Woyczynski, Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992).Google Scholar
  9. 9.
    S.A. Molchanov, D. Surgailis and W.A. Woyczynski, Hyperbolic asymptotics in Burgers Turbulence and extremal processes, Comm. Math. Phys., 1–27 (1994).Google Scholar
  10. 10.
    P.J.E. Peebles, The Large-Scale Structure of the Universe, (1980) Princeton University Press, Princeton, N.J.Google Scholar
  11. 11.
    S.F. Shandarin and B.Z. Zeldovich, Turbulence (1989), intermittency, structures in a self-gravitating medium: the large scale structure of the universe, Rev. Modern Phys., 61, 185–220.Google Scholar
  12. 12.
    Surgailis D., Woyczynski W.A. (1993), Long range predictions and scaling limit for statistical solutions of the Burgers` equation, in Nonlinear Waves and Weak Turbulence, Birkhäuser, Boston, pp. 313–338.Google Scholar
  13. 13.
    Surgailis D., Woyczynski W.A. (1994), Scaling limits of the solution of the Burgers` equation with singular Gaussian initial data, in Chaos Expansions, Multiple Wiener-Ito Integrals and Their Applications, C. Houdre and Perez-Abren, Eds. CRC Press, pp. 145–162.Google Scholar
  14. 14.
    Vergassola M., Dubruille B., Frisch U. and Noulles A. (1994), Burgers' equation, devil's staircases and the mass distribution for the large-scale structures, Astron. et Astrophys., to appear.Google Scholar
  15. 15.
    Weinberg D.H., Gunn J.E. (1990), Large-scale structure and the adhesion approximation, Monthly Not. Royal Astronom. Soc. 247, 260–286.Google Scholar
  16. 16.
    Woyczynski W.A., Stochastic Burgers` flows, in Nonlinear Waves and Weak Turbulence with Applications in Oceanography and Condensed Matter Physics, N. Fitzmaurice et al., eds. 279–312. Birkhauser, Boston (1993).Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • W. A. Woyczynskil
    • 1
    • 2
  • Yiming Hu
    • 2
  1. 1.Department of StatisticsCase Western Reserve UniversityCleveland
  2. 2.Center for Stochastic and Chaotic Processes in Science and TechnologyCase Western Reserve UniversityCleveland

Personalised recommendations