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

• 成果类型: Article
• 署名作者: ALBEVERIO, S; MOLCHANOV, SA; SURGAILIS, D
• 署名单位: Vilnius University; University of North Carolina; University of North Carolina Charlotte
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/BF01268990
• 发表日期: 1994
• 页码: 457-484
• 关键词: CENTRAL LIMIT-THEOREMS; large-scale structure; gaussian fields; turbulence; functionals; BEHAVIOR

摘要

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 held with ''high'' amplitudes) which reveals an intermittent large time behaviour, with the velocity v(t,x) determined by the position of the largest initial fluctuation (discounted by the heat kernel g(t,x, . )) in a neighborhood of x. The asymptotics of such local maximum as t --> infinity 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 of t-local maxima, and show the weak convergence of the suitably rescaled process to a non-trivial limit as t --> infinity.