Discrete- and continuous-time random walks in 1D Lévy random medium

Abstract

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.

15 pages, final version to be published in the proceedings of the conference "The Legacy of Carlo Cercignani: from Kinetic Theory to Turbulence Modeling"