Area coverage of radial Lévy flights with periodic boundary conditions

We consider the area coverage of radial Lévy flights in a finite square area with periodic boundary conditions. From simulations we show how the fractal path dimension d(f) and thus the degree of area coverage depends on the number of steps of the trajectory, the size of the area, and the resolution of the applied box counting algorithm. For sufficiently long trajectories and not too high resolution, the fractal dimension returned by the box counting method equals two, and in that sense the Lévy flight fully covers the area. Otherwise, the determined fractal dimension equals the stable index of the distribution of jump lengths of the Lévy flight. We provide mathematical expressions for the turnover between these two scaling regimes. As complementary methods to analyze confined Lévy flights we investigate fractional order moments of the position for which we also provide scaling arguments. Finally, we study the time evolution of the probability density function and the first passage time density of Lévy flights in a square area. Our findings are of interest for a general understanding of Lévy flights as well as for the analysis of recorded trajectories of animals searching for food or for human motion patterns.