Some Potential Analysis On Right Processes And Lévy Processes

In this thesis, we mainly discuss some potential theory in the framework of Markov processes. Firstly, we study the properties of invariant sets and absorbing sets, introduce the concept of a-excessive function, a-excessive measure, a-recurrent and a-transient for right processes, where a < 0, and give a thorough investigation. Secondly we study the properties of moment generating functions of probability measures, calculate its sub-differential by the convex analysis, use it to characterize the quasi-symmetric probability. Particularly, if the probability measure u is symmetric or if the mean value is 0, then u is quasi-symmetric. Thirdly, we study the a-invariant Radon measures for random walks and Levy processes and give some nice ratio limit theorems that is relative to the a-invariant Radon measures. Fourthly we use the infimum of the moment generating function to study the a-transient and a-recurrent for random walks and Levy processes, give a Dichotomy theorem for not one-sided processes and prove that the process X is quasi-symmetric if and only if X is not a-recurrent for all a < 0 which is the probability explanation of quasi-symmetry. We also study some relationship between random walks and Levy processes. Finally, we give the concept of polynomial recurrent and polynomial transient for Levy processes and give a necessity condition for polynomial recurrence Levy processes which is also a sufficient condition for symmetric Levy processes.