Itô excursion theory for self-similar Markov processes
A standard Markov process on \(([0, \infty), {\mathcal B} [0, \infty))\) with transition function \(P_ t (x,A)\) is considered. It is assumed that the process is \(\alpha\)-selfsimilar, i.e. for some \(\alpha > 0\), \(P_{at} (a^ \alpha x, a^ \alpha A) = P_ t (x,A)\), and that the process is killed when hitting 0. The existence of an \(\alpha\)-selfsimilar extension of the process to \([0, \infty)\) is examined. It is proved that under certain conditions the extension exists and ``either leaves 0 continuously (a.s) or (a.s.) jumps from 0 to \((0, \infty)\) according to the `jumping in' measure \(\eta (dx) = dx/x^{\beta + 1}\), \(\beta > 0\).'' Applications of the result to the diffusion process and to the ``reflecting barrier process'' introduced by S. Watanabe are given.
scientific article; zbMATH DE number 681374