Fractional Lévy motions and related processes

Abstract

This paper investigates some properties of a class of random motions called fractional Lévy motions (FLMs) and their fractal time extension. FLM identifies with fractional Brownian motion (FBM) sampled in fractal Lévy time. This two parameter class of processes borrows hyperbolic temporal dependence to FBM and `heavy-tailedness' to Lévy flights or motions (LM). It is shown that there exists a family of FLMs which shares with standard Brownian motion (BM) the (strict) diffusivity property that the dispersion (measured in terms of quantiles) grows as time raised to the power ½. Processes from this class are critical in that they separate both sub/superdiffusive FLMs and finite/infinite-variance motions. Related Ornstein-Uhlenbeck and multiplicative processes are also briefly investigated. FLMs are self-similar but not Lévy-stable, in sharp contrast to Brownian (whether fractional or not) and standard or fractional stable (in the Taqqu-Wolpert sense) Lévy motions (FSMs). Stable processes are self-similar as a result of Lévy stability. However, the converse is false; there are motions with stationary increments which are self-similar but not stable: the fractional Lévy motion is one of them, in the pure jump process class. It turns out that all processes discussed so far (BM, FBM, LM, FLM, FSM) are self-similar with stationary increments. We finally introduce a natural one-parameter -family of `fractional' processes for which a weaker notion of self-similarity seems to hold, i.e. self-similarity of the unidimensional distributions. It is fractal time Brownian motion (FT-BM). Such a process is obtained as a weak limit of fractal time random walk models, with (0,1) the tail exponent of the waiting times; FT-BM identifies with Brownian motion now sampled in fractal inverse Lévy time. This construction extends to FBM, LM and FLM: we therefore introduce and study FT-FBM, FT-LM and FT-FLM which are the fractal time extensions of FBM, LM and FLM. These processes are not strongly self-similar, nor stable, nor do they have stationary increments.