Random walks in a one-dimensional Lévy random environment

Abstract

We consider a generalization of a one-dimensional stochastic process known in the physical literature as Lévy-Lorentz gas. The process describes the motion of a particle on the real line in the presence of a random array of marked points, whose nearest-neighbor distances are i.i.d. and long-tailed (with finite mean but possibly infinite variance). The motion is a continuous-time, constant-speed interpolation of a symmetric random walk on the marked points. We first study the quenched random walk on the point process, proving the CLT and the convergence of all the accordingly rescaled moments. Then we derive the quenched and annealed CLTs for the continuous-time process.

Final version to be published in J. Stat. Phys. 23 pages. (Changes from v1: Theorem 2.4 and Corollary 2.6 have been removed.)

References in corpus (7)

• Lévy walks
• Nonalgebraic length dependence of transmission through a chain of barriers with a Levy spacing distribution
• Typicality of recurrence for Lorentz gases
• Scattering lengths and universality in superdiffusive Lévy materials
• Diffusivity in one-dimensional generalized Mott variable-range hopping models
• Transport properties of Lévy walks: an analysis in terms of multistate processes
• Central Limit Theorem and recurrence for random walks in bistochastic random environments

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