Random Walks in a One-Dimensional Lévy Random Environment

We consider a generalization of a one-dimensional stochastic process known in the physical literature as Lévy-Lorentz gas. The process describes the motion of a particle on the real line in the presence of a random array of marked points, whose nearest-neighbor distances are i.i.d. and long-tailed. The motion is a constant-speed interpolation of a symmetric random walk on the marked points. We study the quenched version of this process under the hypothesis that the distance between two neighboring marked points has finite mean—but possibly infinite variance—and prove the CLT and the convergence of all the accordingly rescaled moments. Thus, contrary to what is believed to hold for the annealed process, the quenched process is truly diffusive.