A stochastic generalization of renormalization-group transformation for continuous-time random walk processes is proposed. The renormalization consists in replacing the jump events from a randomly sized cluster by a single renormalized (i.e., overall) jump. The clustering of the jumps, followed by the corresponding transformation of the interjump time intervals, yields a new class of coupled continuous-time random walks which, applied to modeling of relaxation, lead to the general power-law properties usually fitted with the empirical Havriliak-Negami function.
In this paper we obtain the scaling limit of a multidimensional Lévy walk and describe the detailed structure of the limiting process. The scaling limit is a subordinated α-stable Lévy motion with the parent process and subordinator being strongly dependent processes. The corresponding Langevin picture is derived. We also introduce a useful method of simulating Lévy walks with a predefined spectral measure, which controls the direction of each jump. Our approach can be applied in the analysis of real-life data-we are able to recover the spectral measure from the data and obtain the full characterization of a Lévy walk. We also give examples of some useful spectral measures, which cover a large class of possible scenarios in the modeling of real-life phenomena.
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