Abstract
We study a random walk on a point process given by an ordered array of points (ωk,k∈Z) on the real line. The distances ωk+1−ωk are i.i.d. random variables in the domain of attraction of a β-stable law, with β∈(0,1)∪(1,2). The random walk has i.i.d. jumps such that the transition probabilities between ωk and ωℓ depend on ℓ−k and are given by the distribution of a Z-valued random variable in the domain of attraction of an α-stable law, with α∈(0,1)∪(1,2). Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters α and β, we prove the annealed functional limit theorem for the suitably rescaled process, 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.
Highlights
The expression ‘Lévy random medium’ indicates a stochastic point process, in some space, where the distances between nearby points have heavy-tailed distributions
The random medium that we consider in this paper is perhaps the most natural choice for a Lévy random medium in the real line: a sequence of random points ω =, where ω0 = 0 and the nearest-neighbor distances ζk = ωk+1 − ωk are i.i.d. variables in the normal domain of attraction of a β-stable variable, with β ∈ (0, 1) ∪ (1, 2)
In all cases we prove the optimal, or at least morally optimal, functional limit theorem, meaning that we show distributional convergence of the process with respect to (w.r.t.) the strongest Skorokhod topology that applies there
Summary
The expression ‘Lévy random medium’ indicates a stochastic point process, in some space, where the distances between nearby points have heavy-tailed distributions. Y can be thought of as the limit of a continuous-time random walk with resting times on the points ωk, when the ratio between the speed of the walker and the typical resting time diverges This can be used to model a variety of situations where a given dynamics is very fast compared to its “decision times”, e.g., electronic signal on a network whose nodes act as relatively slow processing stations; human mobility (assuming, as is often the case, that resting times are substantially longer than travel times); etc. This particular model aside, there is no lack of general motivation for the study of random walks on points processes, EJP 26 (2021), paper 57.
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