On exponential functionals, harmonic potential measures and undershoots of subordinators

Abstract

We establish a link between the distribution of an exponential functional I and the undershoots of a subordinator, which is given in terms of the associated harmonic potential measure. This allows us to give a necessary and sufficient condition in terms of the L\'evy measure for the exponential functional to be multiplicative infinitely divisible. We then provide a formula for the moment generating function of an exponential functional $I$ and the so called remainder random variable $R$ associated to it. We provide a realization of the remainder random variable $R$ as an infinite product involving independent last position random variables of the subordinator. Some properties of harmonic measures are obtained and some examples are provided.