L\'evy Processes and L\'evy White Noise as Tempered Distributions

摘要

We identify a necessary and sufficient condition for a L\'evy white noise tobe a tempered distribution. More precisely, we show that if the L\'evy measureassociated with this noise has a positive absolute moment, then the L\'evywhite noise almost surely takes values in the space of tempered distributions.If the L\'evy measure does not have a positive absolute moment of any order,then the event on which the L\'evy white noise is a tempered distribution hasprobability zero.