Collaborative Research: Stochastic Methods for Fractional Partial Differential Equations

布朗运动是一种扩散的随机模型，是简单随机游走的尺度极限。该随机过程的概率密度解经典扩散方程。连续时间随机游走是一种服从于更新过程的简单随机游走，在物理学中用于模拟异常扩散。简单随机游动的增量表示粒子跳跃，更新epoch表示粒子跳跃次数。无限方差粒子跳跃导致超扩散，在这种情况下，粒子云的扩散速度比经典模型预测的要快。无限的平均等待时间导致亚扩散。连续时间随机漫步的尺度极限是密度解分数阶偏微分方程的随机过程。无限方差粒子跳跃导致算子稳定的Levy运动，而无限平均等待时间导致算子从属于一个逆稳定的从属。分数阶导数是稳定连续卷积半群的产生器。由这笔拨款资助的研究正在为这一物理理论建立健全的数学基础，并寻求在污染物运输问题上的实际应用。本研究的目标包括跳跃大小与等待时间之间可能存在依赖关系的连续时间随机行走的极限理论，扩展到更现实的多维跳跃向量，通过矩阵缩放来允许不同的粒子在每个坐标上的扩散速率，在空间中并行开发多尺度分数阶导数算子，分析分数扩散的基本物理基础，阐明模型参数的物理意义，开发有用的参数估计统计方法，分数阶偏微分方程的数值方法，以及将这些方法应用于实验室和现场实验的真实数据，以及涉及多孔介质和裂缝流动的补救工作。山间溪流中污染物的移动需要很长的时间尺度。大多数会很快离开，但一小部分会被困在漩涡中几分钟或几天。另一部分可能会进入河床下相对静止的水中，最后的分子可能需要几个月或几年的时间才能完全消失。地下水中污染物的移动在类似的时间尺度范围内扩散。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运动和传播的精确模型。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质运动的现实模型。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.