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Recurrent Extensions of Real-Valued Self-Similar Markov Processes

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Abstract

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.

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Authors and Affiliations

  1. Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico Norte, Tablaje Cat. 13615, Colonia Chuburná Hidalgo Inn, Mérida, Yucatán, México

    H. Pantí

  2. Centro de Investigación en Matemáticas, Calle Jalisco S/N. Col Valenciana, A.P. 402, C.P. 36000, Guanajuato, Guanajuato, México

    J. C. Pardo & V. M. Rivero

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  1. H. Pantí
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  2. J. C. Pardo
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  3. V. M. Rivero
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Correspondence to H. Pantí.

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Pantí, H., Pardo, J.C. & Rivero, V.M. Recurrent Extensions of Real-Valued Self-Similar Markov Processes. Potential Anal 53, 899–920 (2020). https://doi.org/10.1007/s11118-019-09791-x

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