Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment

Motivated by applications to stochastic processes in random environment, we study the asymptotic behaviour of the expectation of functionals $F$ of exponential functionals of Lévy processes, where $F$ is non increasing and with at least a polynomial decay at infinity. We find five different regimes that depend on the shape of the Laplace exponent of the Lévy process under consideration. Our proof relies on a discretisation of the exponential functional and is closely related to the behaviour of functionals of semi-direct products of random variables. We apply this result to three questions associated to stochastic processes in random environment. We first consider the asymptotic behaviour of extinction and explosion for stable continuous state branching processes in a Lévy random environment. Secondly, we focus on the asymptotic behaviour of the mean of a population model with competition in a Lévy random environment and finally, we study the tail behaviour of the maximum of a diffusion process in a Lévy random environment.