Stochastic Models for Anomalous Diffusion

经典扩散是分子碰撞作用下物质扩散的数学模型。同一模型还描述了各种色散现象，其中扩散是由于其他机制，如速度对比、跳变和捕获。当扩散速度比经典模型预测的快（超扩散）或慢（亚扩散）时，就会发生异常扩散。污染物在空气和水中的扩散、生物物种的扩散、分子通过细胞膜的运动以及股票价格的波动都可以看到这种现象。随机方法识别确定性扩散方程背后的随机运动。他们描述了异常扩散方程的物理原理，并通过粒子跟踪的方法促进了数值解，其中大量的代理通过时间和空间来模拟随机物理模型。幂律静止时间导致反常扩散方程的分数阶时间导数，而幂律运动导致空间的分数阶导数。但是，理论上幂律的适用范围受到技术条件的限制。提出的研究将改进和扩展异常扩散的随机模型，放宽这些技术条件，以允许任意幂律指数。它还将包括替代随机模型，通过使用三角阵列限制和马尔可夫过程方法，导致广泛的运动和传播的时空伪微分方程模型。异常扩散方程背后的随机过程模型是连续时间随机游走的标度极限，其中随机等待时间介入随机运动之间。尺度限制是马尔可夫过程服从于非马尔可夫命中时间过程。由此产生的模型将为地球物理学、生物学和金融学的重要应用提供坚实的基础。生物或化学制剂一旦释放到空气或水中，就会移动和扩散。拟议的研究将允许对这些污染物的扩散进行更准确的建模。先前的研究记录了空气和水中的异常传播。快速扩散会导致污染物比预期更早到达下游。准确预测这一风险因素对于保护饮用水源、正确评估通过空气释放的生物或化学攻击的风险以及设计一个安全的核废料储存库都很重要。在治理水污染的努力中，已经观察到缓慢释放是一个相关的问题。改进的模式会增强国家？美国为超级基金整治工作制定预算的能力。在药物跨越细胞边界的运动、股票市场价格的变化和高级复合材料形成过程中分子的迁移中也观察到异常扩散。拟议的研究将为更精确的医疗药物输送设计系统，更好的退休投资组合管理以及使用复合材料改进制造奠定基础。

Classical diffusion is a mathematical model for the spread of agents due to molecular collisions. The same model also describes various dispersion phenomena, where the spreading is due to other mechanisms such as velocity contrasts, hopping, and trapping. Anomalous diffusion occurs when the rate of spreading is either faster (super-diffusion) or slower (subdiffusion) than the classical model predicts. This phenomenon is seen in the spread of contaminants in air and water, the dispersion of biological species, the movement of molecules through cell membranes, and the fluctuations of stock prices. Stochastic methods identify the random motions behind the deterministic diffusion equations. They describe the physical principles that underlie the anomalous diffusion equations, and facilitate numerical solution by the method of particle tracking, where a large number of agents are followed through time and space to mimic the stochastic physical model. Power law resting times lead to fractional time derivatives in the anomalous diffusion equations, while power law movements lead to fractional derivatives in space. However, the range of power laws is restricted by technical conditions in the theory. The proposed research will refine and extend the stochastic models of anomalous diffusion, relaxing these technical conditions to allow an arbitrary power law index. It will also encompass alternative stochastic models that lead to a wide range of space-time pseudo-differential equation models for movement and spreading, by using triangular array limits and Markov process methods. The stochastic process models behind the anomalous diffusion equations are scaling limits of continuous time random walks, where random waiting times intervene between random motions. The scaling limits are Markov processes subordinated to non-Markovian hitting time processes. The resulting models should provide a sound basis for important applications in geophysics, biology, and finance. A biological or chemical agent, once released into the air or water, will move and spread. The proposed research will allow a more accurate modeling of the spreading of these contaminants. Previous research has documented anomalous spreading in both air and water. Fast spreading can cause pollutants to arrive downstream earlier than expected. Accurate prediction of this risk factor is important for protecting sources of drinking water, for properly assessing the risk from a biological or chemical attack via airborne release, and for designing a safe repository for nuclear waste. Slow release is a related issue that has been observed in efforts to clean up water pollution. Improved models will enhance the nation?s ability to budget for superfund remediation efforts. Anomalous spreading is also observed in the movement of drugs across cell boundaries, stock market price changes, and the migration of molecules in the formation of advanced composite materials. The proposed research will build a foundation for more accurate medical drug delivery design systems, better management of retirement portfolios, and improved manufacturing using composite materials.