Problems in self-similar Markov processes

近几十年来，稳定的Lévy过程和自相似马尔可夫过程之间的相互作用引起了人们的极大兴趣，这也是Lévy过程领域研究的继续动力。特别地，考虑了用条件过程的极限过程产生的Doob h变换刻画马尔可夫过程的规律的问题。作为这个项目的一部分，我们的目标是描述有条件地连续击中d维单位球面的马尔可夫加性过程，建立在现有工作的基础上，这些工作包括使稳定过程连续地接近单位球面。有了这一点，我们将继续发展暂态自相似马尔可夫过程的Williams路径分解，它将过程的路径分解成条件段，我们将对其进行完整的描述。在此过程中，将有必要建立马尔可夫加性过程的几个新的涨落恒等式，这些恒等式本身将构成例如论文中的一章的基础。

The interplay between stable Lévy processes and self-similar Markov processes has attracted much interest in recent decades, and is what continues to motivate research in the field of Lévy processes. In particular, the problem of characterising the laws of Markov processes via Doob h-transforms which arise from limiting procedures of conditioned processes has been considered. As part of this project, we aim to describe the Markov Additive Process conditioned to continuously hit the unit sphere in d dimensions, building upon existing work which involved conditioning the stable process to continuously approach the unit sphere. With this in hand, we will proceed to develop a Williams type path decomposition for transient self-similar Markov processes which breaks the path of the process into conditioned segments and for which we will give a full description. En route, it will be necessary to develop several new fluctuation identities for Markov Additive Processes, which, themselves will form the basis of e.g. a chapter in the thesis.