From Isomorphism Identities to Levy measures of Non-negative Infinitely Divisible Processes

In this dissertation, we examine the L ́evy measures of non-negative infinitely divisible processes. For a non-negative infinitely divisible process with no drift, L ́evy measure is the single most important factor characterizing the process. Understanding the structure of L ́evy measure can give more insight about the behavior of the process. However, it is not always easy to describe the L ́evy measure of an infinitely divisible process. The descriptions of L ́evy measures of squared Bessel processes proposed by Pitman and Yor are examples. It requires deep knowledge from the Ray-Knight theorems and Itoˆ excursion laws to interpret these descriptions. We use isomorphism identities as the main tool to describe the L ́evy measure of a non-negative infinitely divisible process. The isomorphism identities that we are interested in connect every non-negative infinitely divisible process to the family of its random translations. It turns out that the L ́evy measure of a non-negative infinitely divisible process can be written in term of the laws of its random translations. More precisely, we manage to write the L ́evy measure as a series of other L ́evy measures which are written in term of the law of random translations. The special technique that we develop to ensure the condition distribution of a non-negative infinitely divisible process being consistently well defined enables us to find the laws corresponds to the component L ́evy measures in the series. Using this method, we have an alternative way to describe the L ́evy measures of squared Bessel processes.