AbstractThe aim of the thesis is to characterise new ways of path-conditioning of a d-dimensional isotropic stable Lévy process and consider their time-reversed paths. In our analysis, we use both of classical potential theory approach and recently-developed methods around the theory of self- similar Markov processes. In doing so, we have the opportunity to consider the role of certain harmonic/excessive functions that have not been previously studied.
In the first part of the thesis, Chapter 3, we consider an oscillatory conditioned attraction of the stable Lévy process to a subset of the unit sphere or a hyperplane. We characterise the hitting distribution as well as the time-reversed process of this conditioned processes in the sense of Hunt-Nagasawa duality for Markov processes. The resulting time-reversed processes have the same distribution as the unconditioned stable Lévy process itself when issued from the corresponding subset of the unit sphere or an hyperplane.
In the second part, Chapter 4, we condition the stable Lévy process to remain either outside or inside of the sphere and we characterise the same conditioned attraction to a subset of the unit sphere. The methods of the previous chapter are not applicable in this case. We use instead recent developments in the representation of d-dimensional isotropic stable Lévy processes as a self-similar Markov processes. In particular, we use a characterisation of the point of closest/furthest reach from the unit sphere for the stable Lévy process. As in the first part, we characterise the hitting distribution as well as the time-reversed process of the newly conditioned processes via Hunt-Nagasawa duality. The resulting time-reversed processes have the law of stable Lévy processes conditioned to stay away from the unit sphere and the process conditioned to stay inside the unit sphere and continuously absorbed at the origin correspondingly. We also extend the conditioned processes as well as its time-reversed processes to be issued from the boundary of the domain in which they live.
Finally, we would like point out that the methodology that we use in the Chapter 4 was only possible for a subset of the unit sphere and we could not extend to the setting of the hyperplane. The reason for this is that we use the law of the point of closest reach from the unit sphere to define the conditioning. There is no such result for the case of an hyperplane so far. Thus, for further work, we aim to characterise one-sided attraction to a subset of an hyperplane.
|Date of Award||22 Jun 2022|
|Supervisor||Andreas Kyprianou (Supervisor), Sandra Palau Calderon (Supervisor) & Marcel Ortgiese (Supervisor)|