Multi-Scaling Theory and Methods for Random Fields

[803391]本研究的主要重点是数学理论和方法，这些理论和方法直接关系到在各种现代应用中自然产生的乘法结构中的多尺度现象问题，如流域水文学、流体湍流、金融证券市场、自旋玻璃等。其基本的数学成分是一个由乘法级联随机过程定义的时空随机场和一个由随机树图表示的随机网络，以及一个指导进化的方程组。关于这些结构所收集和报告的大部分数据都是以某种数量相对于长度尺度的对数-对数图的形式出现的。这导致引入和改进新类别的自相似空间/时间模型，其缩放结构是从经验观察到的样本实现中推断出来的。因此，我们试图计算指数统计估计的某些大样本（细尺度）极限和控制波动的极限定律。此外，还寻求了级联模型和网络模型的自相似度和缩放指数。然后研究了流过程的标度指数与级联指数和网络指数之间的联系。从输入和网络上相应的结构函数计算中计算极端流的结构函数（即多尺度指数）的理论的前景定义了本研究的前沿。环境科学的一个基本问题是根据当地气候（如降雨）和地形（河网结构、土壤湿度）的数据，从流域确定河流流量的结构（如极端值）。在世界上大多数地区，可用于规划水坝、洪水保险、军事战术等的信息是以遥感的当地气候和地形的形式提供的。数学公式基于守恒方程（质量，动量）的随机系统，该系统通过从遥感数据估计的标度和多标度指数将流量与乘法随机降雨输入和复杂网络地形联系起来。这种类型的结果的一个实际方面是帮助水文学家和工程师将局部观测推断到更大的尺度，并将预测的流量区域化。然而，广泛的数学框架有助于我们理解各种自然随机现象，如流体湍流、随机投资收益率、可再生自然资源分布、自旋玻璃磁体等，这些现象在空间和/或时间上本质上是乘法的。

9803391WaymireThe main focus of this research is on mathematical theory and methods which have a direct bearing on problems involving multiscale phenomena in multiplicative structures that arise naturally in a wide variety of modern applications, such as river basin hydrology, fluid turbulence, financial securities markets, spin glasses etc. The essential mathematical ingredients are a space-time random field defined by a multiplicative cascade random process and a random network represented by a random tree graph, together with a system of equations directing the evolution. Much of the data collected and reported on these structures is in the form of log-log plots of some quantity versus a length scale. This leads to the introduction and refinements of new classes of self-similar spatial/temporal models whose scaling structure is inferred from empirically observed sample realizations. Thus we seek to calculate certain large-sample (fine scale) limits of statistical estimators of exponents and limit laws governing fluctuations. In addition, self-similarities and scaling exponents are sought for the cascade and network models. Then connections between the scaling exponents of the flow processes and those of the cascade and network exponents may be investigated. The prospect of a theory which computes structure functions, (i.e., multiscaling exponents) for extreme flows from corresponding structure function calculations on the inputs and network defines the frontiers of this research.A fundamental problem from environmental science is to determine the structure of river flows (e.g. extremes) from a basin given data on the local climate (e.g. rainfall) and topography (river network structure, soil moisture). In most parts of the world the information available for the planning of dams, flood insurance, military tactics etc. is in the form of remotely sensed local climate and topography. The mathematical formulation is based on a stochastic system of conservation equations (mass, momentum) which relate the flows to the multiplicative stochastic rainfall inputs and complex network topography via scaling and multiscaling exponents which are estimated from remotely sensed data. One of the practical aspects of results of this type is to assist hydrologists and engineers in extrapolating localized observations to larger scales, and to regionalize predicted flows. However, the broad mathematical framework contributes to our understanding of diverse natural stochastic phenomena such as fluid turbulence, stochastic investment yields, renewable natural resource distributions, spin glass magnets etc., which are intrinsically multiplicative in space and/or time.