Continuous-time random walk between Lévy-spaced targets in the real line

Abstract

We consider a continuous-time random walk which is defined as an interpolation of a random walk on a point process on the real line. The distances between neighboring points of the point process are i.i.d. random variables in the normal domain of attraction of an $\alpha$-stable distribution with $0<\alpha <1$. This is therefore an example of a random walk in a Lévy random medium. Specifically, it is a generalization of a process known in the physical literature as Lévy-Lorentz gas. We prove that the annealed version of the process is superdiffusive with scaling exponent $1/(\alpha +1)$ and identify the limiting process, which is not càdlàg. The proofs are based on the technique of Kesten and Spitzer for random walks in random scenery.

Final version to be published in Stochastic Processes and their Applications. 27 pages

References in corpus (7)

• Lévy walks
• The Single Big Jump Principle in Physical Modelling
• One-Dimensional Stochastic Lévy-Lorentz Gas
• Lévy walks and scaling in quenched disordered media
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Cited by in corpus (6)

• The Single Big Jump Principle in Physical Modelling
• Statistics of occupation time and connection to local properties of non-homogeneous random walks
• Rare events in stochastic processes with sub-exponential distributions and the Big Jump principle
• Fast rare events in exit times distributions of jump processes
• Large fluctuations and transport properties of the Lévy-Lorentz gas
• Ladder costs for random walks in Lévy random media