连续时间随机游动的极限行为及其相关研究

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.

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。

本项目研究了Lévy过程极大逗留点的极限行为，在没有正则变化条件的情况下推广了前人对布朗运动、稳定过程、部分Lévy过程的结果。连续时间随机游动（CTRW）是一个随机等待时间和随机跳跃的复合模型，我们对几类特殊CTRW的极限过程，计算了极限过程的集中概率并研究了其轨道的分形性质。不考虑等待时间，我们研究了模型的特殊情形，复合Poisson分布。我们证明了离散复合Poisson分布的几种刻画的等价性，研究了广义的伪离散复合 Poisson分布及对应的伪无穷可分分布，推广了 Feller 对经典整数值复合 Poisson 分布等价于整数值无穷可分分布的定理。我们将伪复合Poisson分布应用到经典保险模型中得到了相关结果，我们希望将离散复合Poisson分布在数理统计中做更多的应用和推广。