Lévy flights versus Lévy walks in bounded domains

Abstract

Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities are discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. In consequence, well developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flights and Lévy walks models on bounded domains examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time and stationary PDFs. It is demonstrated that similarity of models is affected by the type of boundary conditions and value of the stability index defining asymptotics of the jump length distribution.

15 pages, 13 figures

References in corpus (9)

• Lévy walks
• Fractional Laplacian in Bounded Domains
• Infinite densities for Lévy walks
• Intermediate dynamics between Newton and Langevin
• Asymptotic densities of ballistic Lévy walks
• Harmonic oscillator under Levy noise: Unexpected properties in the phase space
• Transport in a Levy ratchet: Group velocity and distribution spread
• Steady-State Lévy Flights in a Confined Domain
• The Spectrum of the Fractional Laplacian and First Passage Time Statistics

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