多重分形行为的描述综述
A survey on prescription of multifractal behavior
摘要：
多重分形行为已被确定，并在数学上建立了大类的功能，随机过程和措施。多重分形也被观察到许多数据来自地球物理学，湍流，物理学，生物学，仅举几例。因此，开发其标度和多重分形特性符合数据测量结果的数学模型是一个重要的问题。这就提出了几个尚未解决的理论问题：多重分形的规定（即如何建立具有已知奇异谱的数学模型），函数空间的典型行为，以及可能具有多重分形行为的偏微分方程或单偏微分方程的解的存在性。在这次调查中，我们收集了这方面的一些最新结果。献给Alain Arn odo，数据分析小波工具开发的先驱.
英文摘要：
Multifractal behavior has been identified and mathematically established for large classes of functions, stochastic processes and measures. Multifractality has also been observed on many data coming from Geophysics, turbulence, Physics, Biology, to name a few. Developing mathematical models whose scaling and multifractal properties fit those measured on data is thus an important issue. This raises several still unsolved theoretical questions about the prescription of multifractality (i.e. how to build mathematical models with a singularity spectrum known in advance), typical behavior in function spaces, and existence of solutions to PDEs or SPDEs with possible multifractal behavior. In this survey, we gather some of the latest results in this area. Dedicated to Alain Arn{é}odo, pioneer in the development of wavelet tools for data analysis.
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