We study the exponential functional ∫ ∞ 0 e −ξs− dη s of two one-dimensional independent Lévy processes ξ and η, where η is a subordinator. In particular, we derive an integro-differential equation for the density of the exponential functional whenever it exists. Further, we consider the mapping Φ ξ for a fixed Lévy process ξ, which maps the law of η 1 to the law of the corresponding exponential functional ∫ ∞ 0 e −ξs− dη s , and study the behaviour of the range of Φ ξ for varying

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... s of ξ. Moreover, we derive conditions for selfdecomposable distributions and generalized Gamma convolutions to be in the range. On the way we also obtain new characterizations of these classes of distributions.