Asymptotic properties and numerical simulation of multidimensional Lévy walks

• We build a complete asymptotic theory of multidimensional Lévy walks.
• We show that the resulting limiting processes belong to sub-, quasi- and superdiffusion regimes.
• We derive the corresponding fractional diffusion equations.
• We derive the Langevin picture of multidimensional Lévy walks.
• We propose a simulation method for trajectories of the limit process.

In this paper we analyze multidimensional Lévy walks with power-law dependence between waiting times and jumps. We obtain the detailed structure of the scaling limits of such multidimensional processes for all positive values of the power-law exponent. It appears that the scaling limit strongly depends on the value of the power-law exponent and has two possible scenarios: an αα-stable Lévy motion subordinated to a strongly dependent inverse subordinator, or a Brownian motion subordinated to an independent inverse subordinator. Moreover, we derive the mean-squared displacement for the scaling limit processes. Based on these results we conclude that the resulting limiting processes belong to sub-, quasi- and superdiffusion regimes. The corresponding fractional diffusion equation and Langevin picture of considered models are also derived. Theoretical results are illustrated using the proposed numerical scheme for simulation of considered processes.