随机游动是具有悠久研究历史和丰富应用背景的经典课题。考虑随机的跳跃位移和等待时间，随机游动便自然地推广成为连续时间随机游动。由于在物理、水文、金融等领域有着极广的应用，其正规化极限过程往往是非高斯、非马尔科夫过程，具有许多未知而有趣的性质，连续时间随机游动及其极限行为已然成为概率研究的热点。本项目的核心问题即是连续时间随机游动极限过程的刻画和性质。一方面，研究各种情形下极限过程的相关性质，尤其是其样本轨道的分形性质;另一方面，在恰当的不独立假设下寻找新生成的极限过程的刻画，力求揭示连续时间随机游动极限过程与分数阶Cauchy问题之间的密切联系和一般规律。本项目探索研究该领域的新方法和新理论，同时由于连续时间随机游动本身是多个学科的基础模型，本项目的成果可以在相关领域直接应用。因此，本项目的研究将在丰富数学理论的同时，对金融、风险、物理、水文等相关领域有着较大的促进作用。

Random walk is a classic topic with long history and rich background. Continuous time random walk (CTRW) generalizes ordinary random walk by implementing random waiting times between successive random jumps. In recent years, CTRW models have found many applications in physics, hydrology, finance and insurance risk analysis. For example, CTRW has been used in physics to model a variety of phenomena connected with anomalous diffusion. .In general, the scaling limit of CTRW is non-Markovian and non-Gaussian，which raises many interesting and challenging mathematical problems. This project will focus on the characterization and the properties of the scaling limits..Problem 1. Investigate further probabilistic and statistical properties of the scaling limits of CTRW. Especially, we pay more attention to the fractal properties of sample path..Problem 2. Under appropriate dependence structures on the jumps and waiting times, identify the scaling limits. Using the fundamental ideas of Skorokhod, we develop the space-time fractional diffusion equations that govern CTRW scaling limits. This also develops classical solutions for fractional Cauchy problems..Non-trivial modifications of the existing methods or even completely new methods will have to be used in our project. And the results of this project can be applied directly to so many relevant fields. Therefore, this project will not only richen mathematical theory but also promote the related application fields such as finance, risk, physics, hydrology etc.