Abstract
For any two-sided jumping $\alpha$-stable process, where $1 < \alpha < 2$, we find an explicit identity for the law of the first hitting time of the origin. This complements existing work in the symmetric case and the spectrally one-sided case; cf. Yano-Yano-Yor (2009) and Cordero (2010), and Peskir (2008) respectively. We appeal to the Lamperti-Kiu representation of Chaumont-Pant\'i-Rivero (2011) for real-valued self-similar Markov processes. Our main result follows by considering a vector-valued functional equation for the Mellin transform of the integrated exponential Markov additive process in the Lamperti-Kiu representation. We conclude our presentation with some applications.
Highlights
Let X := (Xt)t≥0 be a one-dimensional Lévy process, starting from zero, with lawP
The distribution of T0 is equal to that of I(αξ), and we develop techniques to compute a vector-valued Mellin transform for the exponential function of this Markov additive process
In certain scenarios the distribution of T0 is a very convenient quantity to have, and we consider some applications in section 4: for example, we give an alternative description of the stable process conditioned to avoid zero, and we give some identities in law similar to the result of Bertoin and Yor [7] for the entrance law of a pssMp started at zero
Summary
Let X := (Xt)t≥0 be a one-dimensional Lévy process, starting from zero, with law. P. Manipulating this formula, they derive the Mellin transform of T0 Their proof is rather shorter than ours, but it appears to us that the Markov additive point of view offers a good insight into the structure of real self-similar Markov processes in general, and, for example, will be central to the development of a theory of entrance laws of recurrent extensions of rssMps. In certain scenarios the distribution of T0 is a very convenient quantity to have, and we consider some applications in section 4: for example, we give an alternative description of the stable process conditioned to avoid zero, and we give some identities in law similar to the result of Bertoin and Yor [7] for the entrance law of a pssMp started at zero
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