STRATIFIED STRUCTURE OF THE UNIVERSE AND BURGERS-EQUATION - A PROBABILISTIC APPROACH

The model of the potential turbulence described by the 3-dimensional B urgers' 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 la rge 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 a nd the shot-noise model, In Sect. 3 we construct a probabilistic model of strong initial fluctuations (a zero-range shot-noise held with ''h igh'' amplitudes) which reveals an intermittent large time behaviour, with the velocity v(t,x) determined by the position of the largest ini tial fluctuation (discounted by the heat kernel g(t,x, . )) in a neigh borhood of x. The asymptotics of such local maximum as t --> infinity can be analyzed with the help of the theory of records (Sect. 4). Fina lly, in Sect. 5 we introduce a global definition of a point process of t-local maxima, and show the weak convergence of the suitably rescale d process to a non-trivial limit as t --> infinity.