LEVY MULTIPLICATIVE CHAOS AND STAR SCALE INVARIANT RANDOM MEASURES

• 成果类型: Article
• 署名作者: Rhodes, Remi; Sohier, Julien; Vargas, Vincent
• 署名单位: Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Eindhoven University of Technology; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite PSL; Ecole Normale Superieure (ENS)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP810
• 发表日期: 2014
• 页码: 689-724
• 关键词: iterated random multiplications; multifractal random measures; turbulence; distributions

摘要

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. We obtain an explicit characterization of the structure of these measures, which reflects the constraints imposed by the continuous setting. In particular, we show that the continuous equation enjoys some specific properties that do not appear in the discrete star equation. To that purpose, we define a Levy multiplicative chaos that generalizes the already existing constructions.