Power Levy motion. I. Diffusion

Abstract

Recently introduced and explored, power Brownian motion (PBM) is a versatile generalization of Brownian motion: it is Markovian on the one hand and it displays a variety of anomalous-diffusion behaviors on the other hand. Brownian motion is the universal scaling-limit of finite-variance random walks. Shifting from the finite-variance realm to the infinite-variance realm, the counterpart of Brownian motion is Levy motion: the stable and symmetric Levy process. This pair of papers introduces and explores power Levy motion (PLM), which is to Levy motion what PBM is to Brownian motion. This first part of the pair constructs PLM and explains its emergence and rationale. Taking on a "diffusion perspective," this part addresses the following facets and features of PLM: increments and their Fourier structure, selfsimilarity and Hurst exponent, sub-diffusion and super-diffusion, aging and anti-aging, and Holder exponent. Taking on an "evolution perspective," the second part will continue the investigation of PLM.