  We consider the time evolution of two-dimensional Levy flights in a finite
area with periodic boundary conditions. From simulations we show that the
fractal path dimension d_f and thus the degree of area coverage grows in time
until it reaches the saturation value d_f=2 at sufficiently long times. We also
investigate the time evolution of the probability density function and
associated moments in these boundary conditions. Finally we consider the mean
first passage time as function of the stable index. Our findings are of
interest to assess the ergodic behavior of Levy flights, to estimate their
efficiency as stochastic search mechanisms and to discriminate them from other
types of search processes.
