Lévy random walks with fluctuating step number and multiscale behavior

Abstract

Random walks with step number fluctuations are examined in n dimensions for when step lengths comprising the walk are governed by stable distributions, or by random variables having power-law tails. When the number of steps taken in the walk is large and uncorrelated, the conditions of the Lévy-Gnedenko generalization of the central limit theorem obtain. When the number of steps is correlated, infinitely divisible limiting distributions result that can have Lévy-like behavior in their tails but can exhibit a different power law at small scales. For the special case of individual steps in the walk being Gaussian distributed, the infinitely divisible class of K distributions result. The convergence to limiting distributions is investigated and shown to be ultraslow. Random walks formed from a finite number of steps modify the behavior and naturally produce an inner scale. The single class of distributions derived have as special cases, K distributions, stable distributions, distributions with power-law tails, and those characteristic of high and low frequency cascades. The results are compared with cellular automata simulations that are claimed to be paradigmatic of self-organized critical systems.

Limit theorems for randomly coarse grained continuous-time random walks

“…The inconvenience of this method is that useful, explicit inversion formulas can be provided only under some restrictive assumptions. Another method is based directly on the definition of CTRW as a random walk subordinated to a renewal counting process, which allows using the technique of randomly indexed sequences and limit theorems for stochastic processes (see [4,5,19,20,25,41,42,48,49,66]). In the latter approach, in contrast to the very popu-lar Tauberian analysis of the Fourier-Laplace transform of the total distance, the limiting distribution can be identified precisely and given in an easy-to-follow form, convenient for further applications.…”

Continuous transforming the BTW to the Manna model

“…Such a structure is observed for all the three intermediate models and all system sizes, except for the cases where p is close to its extreme values; in these cases the linear regions become very small and the regimes cannot be distinguished clearly. Such dual power laws have been previously observed in the context of sandpile models, although the origin and interpretation may be different; for example it has been experimentally found that the distance traveled by tracer grains in a rice pile shows two distinct power laws [36], a fact which has also been demonstrated theoretically and by simulation [37,38]. Also in a sandpile model which has some long-ranged connections it is found that the distribution function of avalanche sizes shows two different exponents [39].…”

Runoff generation in karst catchments: multifractal analysis

“…Recently Painter et al (2002) showed that the elementary steps of a particle traveling through a fractured rock formation follow closely the Pareto distribution, which leads to a Lévy stable distribution of the travel time to the outlet of the system (Hopcraft et al, 1999). These findings suggest that at the springs the BTC of the dissolved carbonate, and that of the electric conductivity as well, may follow the Lévy-stable distribution.…”