Asymptotic results for exponential functionals of Lévy processes

The asymptotic behavior of expectations of some exponential functionals of a Lévy process is studied. The key point is the observation that the asymptotics only depend on the sample paths with slowly decreasing local infimum. We give not only the convergence rate but also the expression of the limiting coefficient. The latter is given in terms of some transformations of the Lévy process based on its renewal function. As an application, we give an exact evaluation of the decay rate of the survival probability of a continuous-state branching process in random environment with stable branching mechanism.