Problems in self-similar Markov processes

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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.