Additive Functionals of Markov Processes

加性泛函的研究是概率论的重要组成部分，因为它是用遍历性和递归性等性质来描述随机过程的工具。本课题旨在研究与马尔可夫过程相关的加性泛函的各种极限规律。最近的研究表明，Harris递归马尔可夫过程的加性泛函的渐近幅度是由部分格林函数来度量的。这一进展表明，在更广泛的情况下，还有更多的问题和机构。该项目列出了本研究将研究相关加性泛函的极限规律的四个领域。作为通向一般理论的第一步，研究者将寻找加法泛函的弱律和强律之间的关系。具体地说，研究者将研究扩散过程的可加泛函的极限定理和嵌入在各种范数空间中的局部时的极限定理，以及由相互作用的粒子系统产生的占据时的强大定律。这项研究的意义还在于它与一些重要问题的联系。加性泛函的极限律在计算机仿真和控制理论中得到了广泛的应用。一般而言，可加泛函描述的是随机积累--特别是投资者在市场环境中的资本积累。研究粒子系统产生的加性泛函将有助于更好地理解各种物种的种群增长和迁移，以及某些疾病的传播过程。这样的模型可以被看作是随时间演化的随机系统，本项目是关于这些系统的长期行为的。这项研究可以实现：1)加性泛函极限定律的一般理论的进展；2)加性泛函研究的技术和工具的发展；3)解决一些实际有趣的模型提出的问题。

0102238ChenThe study of additive functionals is an important part of probability theory, as it serves as a tool of describing the stochastic processes in terms of properties like ergodicity and recurrence. This project is to investigate various limit laws for additive functionals associated to Markov processes. The recent study shows that the asymptotic magnitudes of additive functionals of a Harris recurrent Markov process are measured by the partial Green functions. The progress suggests further questions and establishments in broader situations. The project lists four areas in which the limit laws of related additive functionals will be studied under this research. As the first step toward the general theory, the investigator will look for relations between weak and strong laws for additive functionals. Specifically, the investigator will study the limit theorems for additive functionals of diffusion processes and for the local times embedded in various norm spaces, and the strong laws for occupation times arising from interacting particle systems. The significance of this study is also due to its connections to some important problems. The limit laws for additive functionals have been extensively applied to computer simulation and control theory. In general, additive functionals describe random accumulations --- in particular, the capital accumulations of investors in market environment. The study of additive functionals arising from particle system will lead to better understanding of population growth and migration of various species, and of the spreading process of certain diseases. Such models can be viewed as random system evolving in time and this project is about the long term behaviors of these systems. This research could achieve: 1) progress in the general theory of the limit laws for additive functionals; 2) developments of technologies and tools for study of additive functionals; 3) solutions to the problems raised from some practically interesting models.