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We consider a previously devised model describing Levy random walks (Phys.Rev E 79, 011110; 80, 031148, (2009)). It is demonstrated numerically that thegiven model describes Levy random walks with superdiffusive, ballistic, as wellas superballistic dynamics. Previously only the superdiffusive regime has beenanalyzed. In this model the walker velocity is governed by a nonlinear Langevinequation. Analyzing the crossover from small to large time scales we find thetime scales on which the velocity correlations decay and the walker motionessentially exhibits Levy statistics. Our analysis is based on the analysis ofthe geometric means of walker displacements and allows us to tackle probabilitydensity functions with power-law tails and, correspondingly, divergent moments.

Continuous Markovian model for Levy random walks with superdiffusive and superballistic regimes