Stochastic Models for Anomalous Diffusion

经典扩散是一个关于分子碰撞引起的试剂扩散的数学模型。同一模型还描述了各种扩散现象，其中扩散是由其他机制引起的，例如速度差、跳跃和陷阱。当扩散速度比经典模型预测的更快(超扩散)或更慢(次扩散)时，就会发生异常扩散。这一现象体现在污染物在空气和水中的扩散、生物物种的扩散、分子在细胞膜中的运动以及股票价格的波动中。随机方法识别确定性扩散方程背后的随机运动。它们描述了反常扩散方程背后的物理原理，并通过粒子跟踪的方法促进了数值求解，其中大量的代理通过时间和空间跟踪来模拟随机物理模型。在反常扩散方程中，幂律静止时间导致分数时间导数，而幂规律运动导致空间分数导数。然而，在理论上，幂定律的适用范围受到技术条件的限制。这项拟议的研究将完善和扩展异常扩散的随机模型，放宽这些技术条件，允许任意的幂指数。它还将包括可供选择的随机模型，这些模型通过使用三角阵列极限和马尔可夫过程方法，产生用于运动和扩散的广泛的时空伪微分方程模型。反常扩散方程背后的随机过程模型是连续时间随机游动的标度极限，其中随机等待时间介于随机运动之间。标度极限是服从非马尔可夫命中时间过程的马尔可夫过程。由此产生的模型应该为地球物理、生物学和金融领域的重要应用提供可靠的基础。生物或化学制剂一旦释放到空气或水中，就会移动和传播。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。之前的研究已经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来源、适当评估通过空中排放造成的生物或化学攻击的风险以及设计安全的核废料储存库非常重要。缓慢释放是在清理水污染的努力中观察到的一个相关问题。改进后的机型将增强国家对S超大资金预算补救的能力。在药物跨越细胞边界的移动、股票市场价格的变化以及先进复合材料形成过程中分子的迁移中，也观察到了异常扩散。拟议的研究将为更准确的医疗药物输送设计系统、更好地管理退休投资组合以及改进使用复合材料的制造奠定基础。

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.