Limit theorems for Lévy flights on a 1D Lévy random medium

The purpose of the presentation is to introduce a version of a stochastic process known in the physical literature as Lévy-Lorentz gas and derive laws of large numbers and functional limit theorems for it. The model can be described as follows: we consider a point process omega given by an ordered array of points on the real line. We call omega the random medium. The nearest-neighbour distances between the points are i.i.d variables in the domain of attraction of a beta-stable distribution with beta belonging to (0,1) U (1,2). A random walk Y takes place on the medium according to the following rule. Independently of omega there exists a random walk S on the integers with i.i.d increments in the normal domain of attraction of an alfa-stable distribution with alfa belonging to (0,1) U (1,2). The role of S is to drive the dynamics of Y on the point process omega. For example, if a realization of S is (0,3,-1,…), the process Y starts at the origin of the real line, then jumps to the third point of omega to the right of 0, then to the first point of omega to the left of 0, and so on. Our process of interest is Y. We may describe it as a Lévy flight on a one-dimensional Lévy random medium. For all combinations of the parameters alfa and beta, we prove the annealed functional limit theorem for the suitably rescaled process Y, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.