Probability Theory and Related Fields

, Volume 100, Issue 4, pp 457–484 | Cite as

Stratified structure of the Universe and Burgers' equation—a probabilistic approach

  • Sergio Albeverio
  • Stanislav A. Molchanov
  • Donatas Surgailis


The model of the potential turbulence described by the 3-dimensional Burgers' equation with random initial data was developped by Zeldovich and Shandarin, in order to explain the existing Large Scale Structure of the Universe. Most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results (the “Gaussian scenario”), under suitable moment assumptions on the initial velocity field. These results and some open questions are discussed in Sect. 2, where we concentrate on the Gaussian model and the shot-noise model. In Sect. 3 we construct a probabilistic model of strong initial fluctuations (a zero-range shot-noise field with “high” amplitudes) which reveals an intermittent large time behaviour, with the velocity\(\vec v(t,x)\) determined by the position of the largest initial fluctuation (discounted by the heat kernelg(t,x·)) in a neighborhood ofx. The asymptoties of such local maximum ast→∞ can be analyzed with the help of the theory of records (Sect. 4). Finally, in Sect. 5 we introduce a global definition of a point process oft-local maxima, and show the weak convergence of the suitably rescaled process to a non-trivial limit ast→∞.

Mathematics Subject Classification (1991)

60F05 60H25 60G70 85A40 


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  1. [ABHK1] Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: A stochastic model for the orbits of planets and satellites—an interpretation of Titius-Bode law. Exp. Math.4, 365–373 (1983)Google Scholar
  2. [ABHK2] Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: Reduction of nonlinear problems to Schrödinger or heat equations: formation of Kepler orbits, singular solutions for hydrodynamical equations. In: Albeverio, S. et al. (eds.): Stochastic Aspects of Classical and Quantum Systems, Proceedings Marseille 1983 (Lect. Notes Math., vol. 1109, pp. 189–206). Berlin Heidelberg New York: Springer 1985Google Scholar
  3. [ABHK3] Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: Newtonian diffusions and planets, with a remark on non-standard Dirichlet forms and polymers. In: Truman, A. and Williams, D. (eds): Stochastic analysis and applications (Lect. Notes Math., vol. 1095, pp. 1–24) Berlin Heidelberg New York: Springer 1984Google Scholar
  4. [ABHK4] Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: Diffusions sur une variété Riemannienne: barrieres infranchissables et applications. Astérisque, vol. 132, pp. 181–201, 1985Google Scholar
  5. [ABHKM] Albeverio, S., Blanchard, Ph., Høegh-Krohn, R., Mebkhout, M.: Strata and voids in galactic structures—a probabilistic approach. (BiBoS Preprint no. 154, 1986)Google Scholar
  6. [ArShZ] Arnold, V.I., Shandarin, S.F., Zeldovich, Ya.B.: The large scale structure of the Universe I. General properties. One- and two-dimensional models. Geophys. Astrophys. Fluid Dyn.20, 111–130 (1982)Google Scholar
  7. [B] Billingsley, P.: Convergence of probability measures. New York: Wiley 1975Google Scholar
  8. [BiGTe] Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. Cambridge: Cambridge University Press 1987Google Scholar
  9. [BrMa] Breuer, P., Major, P.: Central limit theorem for non-linear functionals of Gaussian fields. J. Multivariate Anal.13, 425–441 (1983)Google Scholar
  10. [Bu] Bulinskii, A.V.: CLT for families of integral functionals arising in solving multidimensional Burgers' equation. In: Grigelionis, B. et al. (eds.): Probability theory and mathematical statistics. Proceedings of the 5th Vilnius Conference, vol. 1, pp. 207–216. Utrecht Vilnius: VSP-Mokslas 1990Google Scholar
  11. [BuM] Bulinskii, A.V.: Molchanov, S.A.: Asymptotical normality of a solution of Burgers' equation with random initial data. Theory Probab. Appl.36, 217–235 (1991)Google Scholar
  12. [Da] Darling, D.A.: The influence of the maximum term in the addition of independent random variables. Trans. Am. Math. Soc.73, 95–107 (1952)Google Scholar
  13. [Do] Dobrushin, R.L.: Gaussian and their subordinated self-similar generalized random fields. Ann. Probab.7, 1–28 (1979)Google Scholar
  14. [DoMa] Dobrushin, R.L., Major, P.: Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheor. Verw. Geb.50, 27–52 (1979)Google Scholar
  15. [FSuWo] Funaki, T., Surgailis, D., Woyczynski, W.A.: Gibbs-Cox random fields and Burgers' turbulence. Ann. Appl. Probab. (to appear)Google Scholar
  16. [G] Galambos, J.: The asymptotic theory of extreme order statistics. Melbourne: Krieger 1987Google Scholar
  17. [GiSu] Giraitis, L., Surgailis, D.: On shot noise processes with long range dependence. In: Grigelionis, B. et al. (eds.): Probability theory and mathematical statistics. Proceedings of the 5th Vilnius Conference, vol. 1, pp. 401–408. Utrecht Vilnius: VSP-Mokslas 1990Google Scholar
  18. [GiMSu] Giraitis, L., Molchanov, S.A., Surgailis, D.: Long memory shot noises and limit theorems with application to Burgers' equation. In: Brillinger, D. et al. (eds.): New directions in time series analysis, Part II (IMA Volumes in Mathematics and Its Applications, vol. 46, pp. 153–176). Berlin Heidelberg New York: Springer 1993Google Scholar
  19. [Gr] Grandell, J.: Doubly stochastic Poisson processes. (Lect. Notes Math. vol. 529) Berlin Heidelberg New York: Springer 1976Google Scholar
  20. [GMaSa] Gurbatov, S.N., Malachov, A.N., Saichev, A.I.: Nonlinear random waves and turbulence in nondispersive media: waves, rays and particles. Manchester New York: Manchester University Press 1991Google Scholar
  21. [GSa1] Gurbatov, S.N., Saichev, A.I.: The degeneracy of one-dimensional acoustic turbulence at large Reynolds numbers. Zh. Eksper. Teoret. Fiz.80, 689–703 (1981) (English translation: Soviet Phys. JETP)Google Scholar
  22. [GSa2] Gurbatov, S.N., Saichev, A.I.: Probability distributions and spectra of potential hydrodynamic turbulence. Izv. Vyssh. Uchebn. Zaved. Radiofiz.27, 456–468 (1984) (English translation: Radiophys. and Quantum Electronics)Google Scholar
  23. [HLØUZh] Holden, H., Lindstrøm, T., Øksendal, B., Ubøe, Zhang, T.-S.: The Burgers equation with a noisy force and the stochastic heat equation. (Preprint)Google Scholar
  24. [K] Kallenberg, O.: Random measures, 4th ed. Berlin London: Akademie-Verlag and Academic Press 1986Google Scholar
  25. [LLiRo] Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremal and related properties of random sequences and processes. Berlin Heidelberg New York: Springer 1983Google Scholar
  26. [LeO] Leonenko, N.N., Orsingher, E.: Limit theorems for solutions of Burgers' equation with Gaussian and non-Gaussian initial conditions. (Preprint)Google Scholar
  27. [LiGP] Livschitz, I.M., Gredeskul, S.A., Pastur, L.A.: An introduction to the theory of disordered systems. Moscow: Nauka 1982Google Scholar
  28. [Ma] Major, P.: Multiple Ito-Wiener integrals. (Lect. Notes Math., vol. 849) Berlin Heidelberg New York: Springer 1981Google Scholar
  29. [MM] Malyshev, V.A., Minlos, R.A.: Gibbs random fields: the method of cluster expansions. Moscow: Nauka 1985Google Scholar
  30. [MaRe] Maller, R.A., Resnick, S.I.: Limiting behaviour of sums and the term of maximum modulus. Proc. London Math. Soc.49, 385–422 (1984)Google Scholar
  31. [Me] Meyer, P.A.: Un cours sur les intégrales stochastiques. In: Séminaire de Probabilités X (Lecture Notes Math., vol. 511, pp. 245–400). Berlin Heidelberg New York: Springer 1976Google Scholar
  32. [N] Nevzorov, V.B.: Records. Theory Probab. Appl.32, 201–228 (1987)Google Scholar
  33. [Ré] Rényi, A.: Théorie des éléments saillants d'une suite d'observations. In: Colloquium on Combinatorial Methods in Probability Theory, pp. 104–117. Aarhus: Aarhus University Press 1962Google Scholar
  34. [Ru] Ruelle, D.: Statistical mechanics: rigorous results. New York: Benjamin 1969Google Scholar
  35. [Ro] Rosenblatt, M.: Limit theorems for Fourier transforms of functionals of Gaussian sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb.55, 123–132 (1981)Google Scholar
  36. [Ro2] Rosenblatt, M.: Scale renormalization and random solutions of the Burgers equation. J. Appl. Probab.24, 328–338 (1987)Google Scholar
  37. [ShDZ] Shandarin, S.F., Doroshkevich, A.G., Zeldovich, Ya.B.: The large scale structure of the Universe. Sov. Phys. Usp.24, 328–338 (1987)Google Scholar
  38. [ShDZ] Shandarin, S.F., Doroshkevich, A.G., Zeldovich, Ya.B.: The large scale structure of the Universe. Sov. Phys. Usp.26, 46–76 (1983)Google Scholar
  39. [ShZ] Shandarin, S.F., Zeldovich, Ya.B.: The large scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium. Rev. Modern Phys.61, 185–220 (1989)Google Scholar
  40. [Si1] Sinai, Ya.G.: Two results concerning asymptotic behaviour of solutions of the Burgers equation with force. J. Stat. Phys.64, 1–12 (1991)Google Scholar
  41. [Si2] Sinai, Ya.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys.148, 601–621 (1992)Google Scholar
  42. [SuWo1] Surgailis, D., Woyczynski, W.A.: Scaling limits of solutions of the Burgers' equation with singular Gaussian initial data. In: Houdré, C., Perez-Abreu, V. (eds.): Chaos expansions, multiple Wiener-Ito integrals and their applications, pp. 145–161 CRC Press 1994Google Scholar
  43. [SuWo2] Surgailis, D., Woyczynski, W.A.: Long range prediction and scaling limit for statistical solutions of the Burgers' equation. In: Fitzmaurice N. et al. (eds.): Nonlinear waves and weak turbulence, with applications in oceanography and condensed matter physics, pp. 313–338. Boston: Birkhäuser 1993Google Scholar
  44. [Wo] Woyczynski, W.A.: Stochastic Burgers' flows.Ibid In:, pp. 279–313. —— 1993Google Scholar
  45. [T] Tata, M.N.: On outstanding values in a sequence of random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.12, 9–20 (1962)Google Scholar
  46. [Z] Zeldovich, Ya.B.: Gravitational instability: an approximate theory for large density perturbation. Astron. Astrophys.5, 84–89 (1970)Google Scholar
  47. [ZMSh] Zeldovich, Ya.B., Mamaev, A.V., Shandarin, S.F.: Laboratory observation of caustics, optical simulation of motion of particles and cosmology. Sov. Phys. Usp.26, 77–83 (1983)Google Scholar
  48. [ZMRS] Zeldovich, Ya.B., Molchanov, S.A., Ruzmaikin, A.A., Sokolov, D.D.: Intermittency, diffusion and generation in a nonstationary random medium. Math. Phys. Rev.7, 3–110 (1988)Google Scholar
  49. [ZN] Zeldovich, Ya.B., Novikov, I.D.: The structure and evolution of the Universe. Moscow: Nauka 1975Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Sergio Albeverio
    • 1
    • 2
  • Stanislav A. Molchanov
    • 3
  • Donatas Surgailis
    • 4
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.SFB 237, BiBoS CERFIMLocarno
  3. 3.Mathematic DepartmentUniversity of North CarolinaCharlotteUSA
  4. 4.Institute of Mathematics and InformaticsVilniusLithuania

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