Non-homogeneous persistent random walks and averaged environment for the L\'evy-Lorentz gas

We consider transport properties for a non-homogeneous persistent random walk, that may be viewed as a mean-field version of the L\'evy-Lorentz gas, namely a 1-d model characterized by a fat polynomial tail of the distribution of scatterers' distance, with parameter $\alpha$. By varying the value of $\alpha$ we have a transition from normal transport to superdiffusion, which we characterize by appropriate continuum limits.