AI 中文总结
本文总结了基于CTRWs的异常扩散随机方法，阐述其标度极限技术、与非局部方程的联系，及涵盖通用非局部演化方程的最新成果。
AI 中文摘要
本文给出了基于连续时间随机游走（CTRWs）及其标度极限的异常扩散随机方法的自洽总结。文中介绍了CTRWs及其与半马尔可夫过程理论的关系，描述了研究CTRWs标度极限的通用技术，并讨论了极限过程的半马尔可夫性质。在此基础上，引入了极限过程与非局部（分数型）方程的联系，还阐述了远超分数方程的最新通用研究成果。实际上，本文提出的理论将非常通用的非局部演化方程作为抽象柯西问题，以及有界域内逐点非局部分数扩散方程都包含在内。
英文摘要
These notes provide a self-consistent summary of the stochastic approach to anomalous diffusion based on Continuous Time Random Walks (CTRWs) and their scaling limits. An introduction to CTRWs and their relationship to the theory of semi-Markov processes is provided. A general technique to study scaling limits of CTRWs is then described, and the semi-Markov property of the limiting processes is discussed. With this at hand, the connection of limit processes with non-local (fractional-type) equations is introduced, and the most recent (and general) contributions, going far beyond fractional equations, are also described. Indeed, the theory presented here includes very general non-local evolution equations as abstract Cauchy problems as well as pointwise non-local fractional diffusion equations in bounded domains.
Comments67 pages, 4 figures
