Ito Excursion Theory for Self-Similar Markov Processes

Abstract

Let $X_t$ be an $\alpha$-self-similar Markov process on $(0, \infty)$ killed when hitting 0. $\alpha$-self-similar extensions of $X(t)$ to $\lbrack 0, \infty)$ are studied via Ito execusion theory (entrance laws). We give a condition that guarantees the existence of an extension, which either leaves 0 continuously (a.s.) or (a.s.) jumps from 0 to $(0, \infty)$ according to the "jumping in" measure $\eta(dx) = dx/x^{\beta+1}$. Two applications are given: the diffusion case and the "reflecting barrier process" of S. Watanabe.