Fractal and lacunary stochastic processes
type: Item
instance of: scholarly article
title: Fractal and lacunary stochastic processes
zbMATH Open document ID: 0587.60060
DOI: 10.1007/BF01012302
author: E. W. Montroll
author: Barry D. Hughes
author: Michael F. Shlesinger
published in: Journal of Statistical Physics
publication date: 1983-01-01T00:00:00Z
review text: Discrete-time random walks simulate diffusion if the single-step probability density function (jump distribution) generating the walk is sufficiently short-ranged. In contrast, walks with long-ranged jump distributions considered in this paper simulate Lévy or stable processes. A one-dimensional walk with a self-similar jump distribution (the Weierstrass random walk) and its higher-dimensional generalizations generate fractal trajectories if certain transience criteria are met and lead to simple analogs of deep results on the Hausdorff-Besicovitch dimension of stable processes. The Weierstrass random walk is lacunary (has gaps in the set of allowed steps) and its characteristic function is Weierstrass' non-differentiable function. Other lacunary random walks with characteristic functions related to Riemann's zeta function and certain number-theoretic functions have very interesting analytic structure.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 3942645
zbMATH Keywords: fractals
zbMATH Keywords: lacunary series
zbMATH Keywords: self-similar jump distribution
zbMATH Keywords: Hausdorff- Besicovitch dimension
MaRDI profile type: Publication
cites work: Lattice dynamics, random walks, and nonintegral effective dimensionality
cites work: Q5624436
cites work: Expected Number of Distinct Sites Visited by a Random Walk with an Infinite Variance
cites work: Random walks with self-similar clusters
cites work: Fractal random walks
cites work: Q5794159
cites work: What is Dimension?
cites work: Sample functions of stable processes
cites work: Riemann's example of a continuous, 'nondifferentiable' function
cites work: Q4074972
cites work: Q5846799
cites work: Application de la théorie des équations intégrales linéaires aux systèmes d'équations différentielles non linéaires
cites work: Memory function approach to nonlinear deterministic systems: An exact linear equation
description: scientific article; zbMATH DE number 3942645