Recurrent extensions of real-valued self-similar Markov processes

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.…”