Collaborative Research: Stochastic Methods for Fractional Partial Differential Equations

布朗运动是简单随机游动的标度极限，是一种随机扩散模型。这个随机过程的概率密度解经典扩散方程。连续时间随机游走（英语：Continuous time random walk）是一种服从更新过程的简单随机游走，在物理学中用来模拟反常扩散。简单随机游动的增量表示粒子的跳跃，更新时期表示粒子的跳跃次数。无限变化的粒子跳跃导致超扩散，其中粒子云的传播速度比经典模型预测的要快。无限的平均等待时间导致次扩散。连续时间随机游动的标度极限是一类随机过程，其密度可解分数阶偏微分方程。无穷方差粒子跳跃导致运营商稳定的Levy运动，而无限的平均等待时间诱导从属于一个逆稳定从属。分数阶导数是稳定连续卷积半群的生成元。由该基金资助的研究正在为这一物理理论建立健全的数学基础，并寻求污染物传输问题的实际应用。本研究的目标包括连续时间随机游动的极限理论，跳跃大小和等待时间之间可能存在依赖关系，扩展到更现实的多维跳跃向量，矩阵缩放允许每个坐标中粒子扩散的不同速率，空间中多尺度分数导数算子的并行开发，分析分数扩散的基本物理基础，以阐明模型参数的物理意义，开发用于参数估计的有用统计方法，分数偏微分方程的数值方法，以及将这些方法应用于来自实验室和现场实验的真实的数据，以及涉及多孔介质和裂缝流的补救努力。 污染物在山溪中的运动发生在很大的时间尺度范围内。大多数会迅速离开，但少量可以在几分钟或几天内被漩涡捕获。另一部分可能会进入河床下相对静止的水中，最后一部分分子可能需要数月或数年才能完全消失。污染物在地下水中的运动在类似的时间尺度范围内传播。我们的数学家和水文地质学家团队使用现代跨学科的研究方法来开发这些污染物的移动和扩散的精确模型。这项工作不仅仅是一项学术工作，还需要建立饮用水供应中污染物和营养物运动的现实模型。这项研究是必要的，因为与实际数据相比，现有的扩散模型大大低估了污染物到达下游的时间和浓度。

Brownian motion, the scaling limit of a simple random walk, is a stochastic model for diffusion. Probability densities for this stochastic process solve the classical diffusion equation. A continuous time random walk is a simple random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Increments of the simple random walk represents particle jumps, and the renewal epochs represent the particle jump times. Infinite variance particle jumps cause superdiffusion, in which a cloud of particles spreads faster than the classical model predicts. Infinite mean waiting times lead to subdiffusion. Scaling limits of continuous time random walks are stochastic processes whose densities solve fractional partial differential equations. Infinite variance particle jumps lead to operator stable Levy motions, while infinite mean waiting times induce subordination to an inverse stable subordinator. Fractional derivatives are generators of stable continuous convolution semigroups. The research funded by this grant is developing a sound mathematical basis for this physical theory, and pursuing practical applications to problems in contaminant transport. The goals of this research include limit theory for continuous time random walks with possible dependence between the jump sizes and the waiting times, extension to more realistic multidimensional jump vectors with matrix scaling to allow different rates of particle spreading in each coordinate, parallel development of multiscaling fractional derivative operators in space, analysis of the fundamental physical basis of fractional diffusion to elucidate the physical meaning of the model parameters, development of useful statistical methods for parameter estimation, numerical methods for fractional partial differential equations, and application of these methods to real data from laboratory and field experiments and remediation efforts involving porous media and fracture flow. Movement of contaminants in a mountain stream takes place over a vast range of time scales. A majority will move away quickly, but a small amount can be caught in eddies for minutes or days. Another fraction may move into the relatively motionless water beneath the streambed, and it may take months or years for the last molecules to disappear completely. The movement of contaminants in underground water spreads over a similar range of time scales. Our team of mathematicians and hydrogeologists uses modern interdisciplinary research methods to develop accurate models for the movement and spread of these contaminants. Much more than an academic exercise, this work is needed for realistic models of contaminant and nutrient movement in drinking water supplies. The research is necessary because existing diffusion models greatly underestimate the time and concentration at which contaminants arrive downstream, when compared to actual data.