Recurrent extensions of real-valued self-similar Markov processes

2018/08/01 by Henry Pantí, Pantí, Henry, Juan Carlos Pardo +3 · 1 citation
Mathematics · #60G18 #60G51 #60G52 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1808.00129

openalex publication_date 2018/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X=(X_t, t≥ 0) 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 (2006) and Rivero (2005, 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.

Citations

• Entrance laws at the origin of self-similar Markov processes in high dimensions

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