In this talk we consider a one-dimensional process in random
environment, also known in the physical literature as Levy-Lorentz gas. The environment is provided by a renewal point process that can be seen as a set of randomly arranged targets, while the process roughly describes the displacement
of a particle moving on the line at constant velocity, and changing direction at the targets position with assigned probability.
We investigate the annealed behavior of this process in the case of inter-distances between targets having infinite mean, and establish, under suitable scaling, a functional limit theorem for the process. In particular we show that, contrary to the finite mean case, the behavior of the motion is super- diffusive with explicit scaling limit related to the Kesten-Spitzer process.
The key element of the proof is indeed a representation of the consecutive "hitting times on the set of targets" as a suitable random walk in random scenery.