On some transformations between positive self-similar Markov processes

A path decomposition at the infimum for positive self-similar Markov processes (pssMp) is obtained. Next, several aspects of the conditioning to hit 0 of a pssMp are studied. Associated to a given pssMp XX, that never hits 0, we construct a pssMp X↓X↓ that hits 0 in a finite time. The latter can be viewed as XX conditioned to hit 0 in a finite time, and we prove that this conditioning is determined by the pre-minimum part of XX. Finally, we provide a method for conditioning a pssMp that hits 0 by a jump to do it continuously.