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.
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Supported by the U.S. Department of Energy.
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Hughes, B.D., Montroll, E.W. & Shlesinger, M.F. Fractal and lacunary stochastic processes. J Stat Phys 30, 273–283 (1983). https://doi.org/10.1007/BF01012302
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DOI: https://doi.org/10.1007/BF01012302