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Itô excursion theory for self-similar Markov processes. (English) Zbl 0810.60067

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.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60J60 Diffusion processes
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