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Stratified structure of the Universe and Burgers' equation—a probabilistic approach
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  • Published: December 1994

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

  • Sergio Albeverio1,2,
  • Stanislav A. Molchanov3 &
  • Donatas Surgailis4 

Probability Theory and Related Fields volume 100, pages 457–484 (1994)Cite this article

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  • 70 Citations

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Summary

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→∞.

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Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, D-44801, Bochum, Germany

    Sergio Albeverio

  2. SFB 237, BiBoS CERFIM, Locarno

    Sergio Albeverio

  3. Mathematic Department, University of North Carolina, 28223-9998, Charlotte, N.C., USA

    Stanislav A. Molchanov

  4. Institute of Mathematics and Informatics, 2600, Vilnius, Lithuania

    Donatas Surgailis

Authors
  1. Sergio Albeverio
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  2. Stanislav A. Molchanov
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  3. Donatas Surgailis
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Albeverio, S., Molchanov, S.A. & Surgailis, D. Stratified structure of the Universe and Burgers' equation—a probabilistic approach. Probab. Th. Rel. Fields 100, 457–484 (1994). https://doi.org/10.1007/BF01268990

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  • Received: 06 September 1993

  • Revised: 20 May 1994

  • Issue Date: December 1994

  • DOI: https://doi.org/10.1007/BF01268990

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Mathematics Subject Classification (1991)

  • 60F05
  • 60H25
  • 60G70
  • 85A40
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