On the Range of Exponential Functionals of Levy Processes

We characterize the support of the law of the exponential functional integral(infinity)(0) e(-xi s-) d eta(s) of two one-dimensional independent Levy processes xi and eta. Further, we study the range of the mapping Phi(xi) for a fixed Levy process xi, which maps the law of eta(1) to the law of the corresponding exponential functional integral(infinity)(0)e(-xi s-) d eta(s). It is shown that the range of this mapping is closed under weak convergence and in the special case of positive distributions several characterizations of laws in the range are given.