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

Abstract

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→∞.