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Stochastic Analysis of Random Multifractal Measures

随机多重分形测量的随机分析

基本信息

批准号：
1811087
负责人：
Thomas Alberts
金额：
$ 12.8万
依托单位：
University of Utah
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811087&HistoricalAwards=false
关键词：
Stochastic Analysis Random Multifractal Measures

项目摘要

Fractals are defined by the repetition or near-repetition of a pattern on infinitely many scales. They appear extensively in nature and are prevalent in human-generated data such as financial time series, sound files, and internet traffic patterns. By their very nature fractals are rough, non-differentiable objects that cannot be studied by the classical mathematical methods of calculus. The more modern tools of multifractal analysis, which quantify the degree of roughness that appears on various scales, can be used instead. It is particularly interesting to apply multifractal analysis to randomly generated objects, which by default tend to exhibit fractal behavior. Random measures, which for example describe the rainfall distribution over a particular state or the distribution of energy in turbulent fluid flows, are a particularly fruitful area for employing multifractal analysis. The results of such an analysis can often be used for prediction and forecasting of the underlying random system. This research project aims to further develop the mathematical foundations for analysis of multifractal properties.The aim of this project is to analyze the multifractal properties of a broad class of random measures that appear in models of statistical mechanics. The focus will be on random pinning models, random polymer models, random measures arising in two-dimensional conformally invariant systems, and spectral measures of random matrices. In all cases, the major quantity of interest will be the multifractal spectrum, which quantifies the amount of mass the random measure assigns around various points in the space. Computation of the multifractal spectrum is closely related to the theory of large deviations for computing the decay of probabilities of rare events. Many of the tools from that theory will be employed in this project, including the notion of Gibbs condition for describing the most likely behavior of the random system when a rare event occurs. As all the random measures listed above can be described using tools from stochastic analysis, specifically the Wiener chaos decomposition, a particular goal of this project will be to incorporate tools from stochastic analysis into this theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

分形的定义是一种模式在无限多个尺度上的重复或近似重复。它们在自然界中广泛存在，并普遍存在于人类生成的数据中，如金融时间序列，声音文件和互联网流量模式。就其本质而言，分形是粗糙的、不可微的物体，不能用微积分的经典数学方法来研究。可以使用更现代的多重分形分析工具来代替，它量化了在各种尺度上出现的粗糙程度。将多重分形分析应用于随机生成的对象特别有趣，这些对象默认情况下倾向于表现出分形行为。随机测量，例如描述降雨分布在一个特定的状态或能量分布在湍流流体流动，是一个特别富有成效的领域，采用多重分形分析。这种分析的结果通常可以用于预测和预报潜在的随机系统。本研究课题的目的是进一步发展多重分形特性分析的数学基础，分析统计力学模型中出现的一类随机测度的多重分形特性。重点将放在随机钉扎模型，随机聚合物模型，随机措施所产生的二维共形不变系统，随机矩阵的频谱措施。在所有情况下，感兴趣的主要量将是多重分形谱，它量化了随机测量在空间中各个点周围分配的质量量。多重分形谱的计算与计算稀有事件概率衰减的大偏差理论密切相关。该理论中的许多工具将在本项目中使用，包括吉布斯条件的概念，用于描述随机系统在罕见事件发生时最可能的行为。由于上面列出的所有随机测量都可以使用随机分析工具进行描述，特别是维纳混沌分解，因此该项目的一个特殊目标是将随机分析工具纳入该理论。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。