Fractal and lacunary stochastic processes
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.
scientific article; zbMATH DE number 3942645