Discrete- and Continuous-Time Random Walks in 1D Lévy Random Medium

A Lévy random medium, in a given space, is a random point process where the distances between points, a.k.a. targets, are long-tailed. Random walks visiting the targets of a Lévy random medium have been used to model many (physical, ecological, social) phenomena that exhibit superdiffusion as the result of interactions between an agent and a sparse, complex environment. In this note we consider the simplest non-trivial Lévy random medium, a sequence of points in the real line with i.i.d. long-tailed distances between consecutive targets. A popular example of a continuous-time random walk in this medium is the so-called Lévy-Lorentz gas. We give an account of a number of recent theorems on generalizations and variations of such model, in discrete and continuous time.