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Stratified structure of the Universe and Burgers’ equation – a probabilistic approach. (English) Zbl 0810.60058

Summary: The model of the potential turbulence described by the 3-dimensional Burgers’ equation with random initial data was developed by Ya. B. Zeldovich and S. F. Shandarin [Rev. Modern Phys. 61, 185-220 (1989)]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 Section 2, where we concentrate on the Gaussian model and the shot-noise model. In Section 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 kernel \(g(t,x, \cdot))\) in a neighborhood of \(x\). The asymptotics of such local maximum as \(t \to \infty\) can be analyzed with the help of the theory of records (Section 4). Finally, in Section 5 we introduce a global definition of a point process of \(t\)-local maxima, and show the weak convergence of the suitably rescaled process to a nontrivial limit as \(t \to \infty\).

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
60F05 Central limit and other weak theorems
60G70 Extreme value theory; extremal stochastic processes
85A40 Astrophysical cosmology
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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) · Zbl 0533.70006
[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 1985
[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 1984 · Zbl 0547.60082
[4] [ABHK4] Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: Diffusions sur une variété Riemannienne: barrieres infranchissables et applications. Astérisque, vol. 132, pp. 181-201, 1985
[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)
[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) · Zbl 0542.76058 · doi:10.1080/03091928208209001
[7] [B] Billingsley, P.: Convergence of probability measures. New York: Wiley 1975 · Zbl 0172.21201
[8] [BiGTe] Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. Cambridge: Cambridge University Press 1987 · Zbl 0617.26001
[9] [BrMa] Breuer, P., Major, P.: Central limit theorem for non-linear functionals of Gaussian fields. J. Multivariate Anal.13, 425-441 (1983) · Zbl 0518.60023 · doi:10.1016/0047-259X(83)90019-2
[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 1990
[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) · Zbl 04508650 · doi:10.1137/1136027
[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) · Zbl 0047.37502 · doi:10.1090/S0002-9947-1952-0048726-0
[13] [Do] Dobrushin, R.L.: Gaussian and their subordinated self-similar generalized random fields. Ann. Probab.7, 1-28 (1979) · Zbl 0392.60039 · doi:10.1214/aop/1176995145
[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) · Zbl 0397.60034 · doi:10.1007/BF00535673
[15] [FSuWo] Funaki, T., Surgailis, D., Woyczynski, W.A.: Gibbs-Cox random fields and Burgers’ turbulence. Ann. Appl. Probab. (to appear) · Zbl 0838.60016
[16] [G] Galambos, J.: The asymptotic theory of extreme order statistics. Melbourne: Krieger 1987 · Zbl 0634.62044
[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 1990 · Zbl 0731.60037
[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 1993 · Zbl 0771.60019
[19] [Gr] Grandell, J.: Doubly stochastic Poisson processes. (Lect. Notes Math. vol. 529) Berlin Heidelberg New York: Springer 1976 · Zbl 0339.60053
[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 1991 · Zbl 0860.76002
[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)
[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)
[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) · Zbl 0804.35158
[24] [K] Kallenberg, O.: Random measures, 4th ed. Berlin London: Akademie-Verlag and Academic Press 1986 · Zbl 0345.60032
[25] [LLiRo] Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremal and related properties of random sequences and processes. Berlin Heidelberg New York: Springer 1983 · Zbl 0518.60021
[26] [LeO] Leonenko, N.N., Orsingher, E.: Limit theorems for solutions of Burgers’ equation with Gaussian and non-Gaussian initial conditions. (Preprint) · Zbl 0853.35139
[27] [LiGP] Livschitz, I.M., Gredeskul, S.A., Pastur, L.A.: An introduction to the theory of disordered systems. Moscow: Nauka 1982
[28] [Ma] Major, P.: Multiple Ito-Wiener integrals. (Lect. Notes Math., vol. 849) Berlin Heidelberg New York: Springer 1981 · Zbl 0451.60002
[29] [MM] Malyshev, V.A., Minlos, R.A.: Gibbs random fields: the method of cluster expansions. Moscow: Nauka 1985 · Zbl 0584.60062
[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) · Zbl 0543.60038 · doi:10.1112/plms/s3-49.3.385
[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 1976
[32] [N] Nevzorov, V.B.: Records. Theory Probab. Appl.32, 201-228 (1987) · Zbl 0677.62044 · doi:10.1137/1132032
[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 1962
[34] [Ru] Ruelle, D.: Statistical mechanics: rigorous results. New York: Benjamin 1969 · Zbl 0177.57301
[35] [Ro] Rosenblatt, M.: Limit theorems for Fourier transforms of functionals of Gaussian sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb.55, 123-132 (1981) · Zbl 0447.60016 · doi:10.1007/BF00535155
[36] [Ro2] Rosenblatt, M.: Scale renormalization and random solutions of the Burgers equation. J. Appl. Probab.24, 328-338 (1987) · Zbl 0624.60071 · doi:10.2307/3214257
[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)
[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) · doi:10.1070/PU1983v026n01ABEH004305
[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) · doi:10.1103/RevModPhys.61.185
[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) · Zbl 0978.35500 · doi:10.1007/BF01057866
[41] [Si2] Sinai, Ya.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys.148, 601-621 (1992) · Zbl 0755.60105 · doi:10.1007/BF02096550
[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 1994 · Zbl 0849.35122
[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 1993
[44] [Wo] Woyczynski, W.A.: Stochastic Burgers’ flows.Ibid In:, pp. 279-313. ?? 1993 · Zbl 0803.76067
[45] [T] Tata, M.N.: On outstanding values in a sequence of random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.12, 9-20 (1962) · Zbl 0188.23902 · doi:10.1007/BF00538520
[46] [Z] Zeldovich, Ya.B.: Gravitational instability: an approximate theory for large density perturbation. Astron. Astrophys.5, 84-89 (1970)
[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) · doi:10.1070/PU1983v026n01ABEH004307
[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)
[49] [ZN] Zeldovich, Ya.B., Novikov, I.D.: The structure and evolution of the Universe. Moscow: Nauka 1975
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