We study a random walk on a point process given by an ordered array of points on the real line. The distances are i.i.d. random variables in the domain of attraction of a β-stable law, with . The random walk has i.i.d. jumps such that the transition probabilities between and depend on and are given by the distribution of a -valued random variable in the domain of attraction of an α-stable law, with . 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.
This work was partly supported by the joint UniBo-UniFi-UniPd project “Stochastic dynamics in disordered media and applications in the sciences”. A. Bianchi is partially supported by the PRIN Grant 20155PAWZB “Large Scale Random Structures” (MIUR, Italy) and by the BIRD project 198239/19 “Stochastic processes and applications to disordered systems” (UniPd). M. Lenci is partially supported by the PRIN Grant 2017S35EHN “Regular and stochastic behaviour in dynamical systems” (MIUR, Italy).
We thank Ward Whitt for discussing with us the issue of the -continuity of the addition map (cf. end of Section 3.4). E. Magnanini thanks the Department of Mathematics of Università di Bologna, to which she was affiliated when most of this work was done.
"Limit theorems for Lévy flights on a 1D Lévy random medium." Electron. J. Probab. 26 1 - 25, 2021. https://doi.org/10.1214/21-EJP626