Additive Functionals of Markov Processes

University Of Tennessee Knoxville, Knoxville TN

Investigators

Abstract

0102238 Chen The 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.