Recurrent Extensions of Real-Valued Self-Similar Markov Processes

Abstract

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.

The Fractional Laplacian with Reflections

“…Kim, Song, and Vondraček [41] explore halfspaces using Dirichlet forms; see also Chen and Song [23]. Recent results for the entire real line can be found in Pantí, Pardo, and Rivero [53] and Iksanov and Pilipenko [37]; see also Chaumont,Pantí,and Rivero [21]. Furthermore, Bobrowski [6] studies resurrections or concatenations on metric graphs via exit laws and resolvents.…”

On Rd-valued multi-self-similar Markov processes

“…This extension defines a bijection between the set of these processes and this of S d−1 ×R-valued Markov additive processes, where S d−1 is the sphere in dimension d. Actually this representation does not provide a complete description of R d -valued self-similar Markov processes since it does not give any information on the existence of an entrance law at 0 or a recurrent extension after the first passage time at 0. Regarding these questions, only the real case has been investigated up to now, see [15], [9], [13] and the references therein.…”

“…Let us emphasize that most of our results would not apply for d = 1. However in the latter case multi-self-similar Markov processes coincide with self-similar Markov processes which have already been extensively studied in the literature, see [12], [4], [9], [15] or [13], for instance.…”

Stable windings at the origin

“…Acknowledgements. We would like to thank Victor Rivero for bringing the computations in [32,29] to our attention and Tomasz Żak for useful comments throughout the paper.…”

“…thereby showing that it is finite, we can appeal to reasoning in [32] and [29]; see also Section 4.4. of [20]. Specifically, following the computations in all of these papers (for which some simple facts in Lemma 12 of [14] concerning the construction of P • 0 will be useful) we easily conclude that,…”