Markov process , Lévy process , Stable process , Self-similar Markov process , Excessive function , Doob-h-transform
Abstract:
In this thesis we consider two classical problems in the theory of Markov processes for the special case of a Lévy process: Conditioning the process to avoid a Borel set B and conditioning the process to hit a Borel set B continuously from the outside. The aim in both settings is to characterise the conditioned process as a Doob-h-transform of the process killed on entering the set B.
In the first setting we use an invariant function to transform the killed process to a non-killed process whose state space is the complement of B. Moreover, this transformed process is connected to the process conditioned to avoid B and the long-time behaviour of the transformed process is analysed. We tackle this problem when B is an interval for two classes of Lévy processes. The first one consists of Lévy processes with finite variance and the second one consists of stable processes. Different techniques are needed to handle both cases.
The second problem is kind of a counterpart of the first one. Here, the aim is to transform the process killed on hitting B to a process which hits the boundary of B continuously before hitting the inner of B. For this we use a harmonic function and show that the underlying h-transformed process indeed hits the boundary of B continuously. Furthermore, we connect the h-transformed process to the process conditioned to be absorbed by B in a meaningful way. This problem is tackled when B is an interval and the Lévy process is a stable process.
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