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  • 1
    Electronic Resource
    Electronic Resource
    Fractal and lacunary stochastic processes (1983)
    Springer
    Journal of statistical physics 30 (1983), S. 273-283 
    Details
    ISSN: 1572-9613
    Keywords: Random walks ; fractals ; stable distributions ; lacunary series
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Discrete-time random walks simulate diffusion if the single-step probability density function (jump distribution) generating the walk is sufficiently shortranged. In contrast, walks with long-ranged jump distributions considered in this paper simulate Lévy or stable processes. A one-dimensional walk with a selfsimilar jump distribution (the Weierstrass random walk) and its higherdimensional 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 numbertheoretic functions have very interesting analytic structure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01012302
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Fractal random walks (1982)
    Springer
    Journal of statistical physics 28 (1982), S. 111-126 
    Details
    ISSN: 1572-9613
    Keywords: random walks ; stochastic processes ; fractals ; scaling ; stable distributions ; nondifferentiable functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider a class of random walks (on lattices and in continuous spaces) having infinite mean-squared displacement per step. The probability distribution functions considered generate fractal self-similar trajectories. The characteristic functions (structure functions) of the walks are nonanalytic functions and satisfy scaling equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01011626
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    Paper (German National Licenses)
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