On the Existence of Recurrent Extensions of Self-similar Markov Processes

Abstract

Let $X= (X_t) _{t \geq 0}$ be a self-similar Markov process with values in the non-negative half-line, such that the state $0$ is a trap. We present a necessary and sufficient condition for the existence of a self-similar recurrent extension of $X$ that leaves $0$ continuously. This condition is expressed in terms of the Lévy process associated with $X$ by the Lamperti transformation.