Asymptotic properties and numerical simulation of multidimensional Levy walks

作者：Magdziarz Marcin^*; Teuerle Marek

来源：Communications in Nonlinear Science & Numerical Simulation, 2015, 20(2): 489-505.

DOI：10.1016/j.cnsns.2014.05.029

• Levy walk
• Anomalous diffusion
• Scaling limits
• Convergence in distribution
• Subordination
• Spectral measure
• Monte Carlo methods

In this paper we analyze multidimensional Levy 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 alpha-stable Levy 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.

• 出版日期2015-2