On the future infimum of positive self-similar Markov processes

摘要

We establish integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes and for increasing self-similar Markov processes at 0 and + oo.. Our proofs are based on the Lamperti representation and time reversal arguments due to Chaumont, L. and Pardo, J,C. (Prepublication (L'universite de Paris 6), 2005), These results extend laws of the iterated logarithm for the future infimum of Bessel processes due to Khoshnevisan, D.., Lewis, T.M, and Li, W.V. (On the future infima of some transient processes, Probability Theory and Related Fields, 99, 337-360, 1994).