Recurrent Extensions of Real-Valued Self-Similar Markov Processes

  • H. PantíEmail author
  • J. C. Pardo
  • V. M. Rivero


Let X = (Xt,t ≥ 0) be a self-similar Markov process taking values in \(\mathbb {R}\) 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 (Electron. Commun. Probab. 11, 230–241 2006) and Rivero (Bernoulli 11, 471–509 2005, 13, 1053–1070 2007) 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.


Real self-similar Markov processes Stable processes Markov additive processes Lamperti–Kiu representation Exponential functional 

Mathematics Subject Classification (2010)

60G52 60G18 60G51 


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Facultad de MatemáticasUniversidad Autónoma de YucatánMéridaMéxico
  2. 2.Centro de Investigación en MatemáticasGuanajuatoMéxico

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