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

Let $ξ=(ξ_t, t\ge 0)$ be a real-valued Lévy process and define its associated exponential functional as follows $I_t(ξ):=\int_0^t \exp\{-ξ_s\}{\rm d} s, \qquad t\ge 0.$ Motivated by important applications to stochastic processes in random environment, we study the asymptotic behaviour of $\mathbb{E}\Big[F\big(I_t(ξ)\big)\Big] \qquad \textrm{as}\qquad t\to \infty,$ where $F$ is a non-increasing function with polynomial decay at infinity and under some exponential moment conditions on $ξ$. In particular, we find five different regimes that depend on the shape of the Laplace exponent of $ξ$. Our proof relies on a discretisation of the exponential functional $I_t(ξ)$ and is closely related to the behaviour of functionals of semi-direct products of random variables. We apply our main 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.

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