Analysis of complex systems related to fractional random fields

该提案将使用新的数学技术来分析复杂系统的行为，这些系统在空间和时间上随机演化，其基本结构由随机场，特别是随机分形决定。随机场是物理学、工程学、环境科学和医学成像中建模时空现象的有用工具。该提案将包括与以下两个领域有关的项目：（一）随机分析;（二）极限定理。 (I)随机分析是从伊藤的开创性工作发展而来的，伊藤的开创性工作介绍了布朗运动的随机积分。该领域包括随机偏微分方程（SPDE）的研究，其目标是为随时间演变的随机现象的行为提供严格的数学解释。该程序将分析受高斯噪声扰动的SPDE，高斯噪声比时空白色噪声更一般，特别是具有时间（或空间）上分数布朗运动的结构。这项研究的动机是最近的兴趣随机分形，这是经常遇到的各种应用。 (II)独立观测值的部分和（或最大值）的行为是 概率论中的许多研究，最好通过不变性原理和强近似来理解。在实践中（例如，在金融数据或地理空间数据的情况下），大多数观测结果在时间或空间上表现出某种依赖结构。观测值之间的依赖结构的存在通常会显著改变这些结果的有效性。该建议的目的是在复杂系统的相关观测的情况下导出新的强逼近结果和不变性原理。特别地，具有平稳新息的线性序列（也称为移动平均序列）构成了一大类平稳序列，其表现出特殊的依赖结构，并且适合于建模金融数据或地理空间数据。

This proposal will use novel mathematical techniques for analyzing the behavior of complex systems which evolve randomly in space and time, whose underlying structure is determined by a random field, and in particular, a random fractal. Random fields are useful tools for modeling spatial-temporal phenomena in physics, engineering, environmental sciences, and medical imaging. The proposal will contain projects related to the following two areas: (I) stochastic analysis; (II) limit theorems. (I) Stochastic analysis has been developed from Ito's seminal work, which introduced the stochastic integral with respect to the Brownian motion. This area includes the study of stochastic partial differential equations (SPDE's), whose goal is to offer a rigorous mathematical explanation for the behaviour of random phenomena which evolve over time. This program will analyze SPDE's perturbed by a Gaussian noise, which is more general than the space-time white noise, and in particular has the structure of the fractional Brownian motion in time (or in space). This study is motivated by the recent interest in random fractals, which are encountered frequently in a variety of applications. (II) The behaviour of the partial sum (or maximum) of independent observations lies at the origin of many investigations in probability theory, and is best understood via the invariance principles and strong approximations. In practice (for instance, in the case of financial data or geo-spatial data), most observations exhibit a certain dependence structure over time or space. The presence of a dependence structure between the observations usually alters significantly the validity of these results. This proposal aims at deriving new strong approximation results and invariance principles in the case of dependent observations arising from complex systems. In particular, linear sequences (also called moving average sequences) with stationary innovations constitute a large class of stationary sequences, which exhibit a special dependence structure and are suitable for modeling financial data or geo-spatial data.