In this paper, we obtain a Lamperti type representation for real-valued self-similar Markov processes, killed at their hitting time of zero. Namely, we represent real-valued self-similar Markov processes as time changed multiplicative invariant processes. Doing so, we complete Kiu's work [Stochastic Process. Appl. 10 (1980) 183-191], following some ideas in Chybiryakov [Stochastic Process. Appl. 116 (2006) 857-872] in order to characterize the underlying processes in this representation. We provide some examples where the characteristics of the underlying processes can be computed explicitly.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ460 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The purpose of this paper is to construct the law of a Lévy process conditioned to avoid zero, under mild technicals conditions, two of them being that the point zero is regular for itself and the Lévy process is not a compound Poisson process. Two constructions are proposed, the first lies on the method of h-transformation, which requires a deep study of the associated excessive function; while in the second it is obtained by conditioning the underlying Lévy process to avoid zero up to an independent exponential time whose parameter tends to 0. The former approach generalizes some of the results obtained by Yano [27] in the symmetric case and recovers some of main results in Yano's work [28], while the latter is reminiscent of [8]. We give some properties of the resulting process and we describe in some detail two examples: alpha stable and spectrally negative Lévy processes. MSC: 60G51 60G18 60J99

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Let X " pX t , t ě 0q be a self-similar Markov process taking values in R such that the state 0 is a trap. In this paper, we present a necessary and sufficient condition for the existence of a self-similar recurrent extension of X that leaves 0 continuously. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. Our results extend those of Fitzsimmons [9] and Rivero ([19], [20]) where the existence and uniqueness of a recurrent extension for positive self similar Markov processes were treated. In particular, we describe the recurrent extension of a stable Lévy process which to the best of our knowledge has not been studied before.

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