First passage time density , Levy flight

Before calculating this first passage time density, we first demonstrate the validity of Eq. (100) by means of a simulation the results of which are shown in Fig. 14. Random jumps with Levy flight jump length statistics are performed, and a particle is removed when it enters a certain interval of width w around the sink in our simulations we found an optimum value w 0.3. As seen in Fig. 14 (note that we plot gtp(t) ) and for analogous results not shown here, relation (100) is satisfied for 1 < a < 2, whereas for larger w, the slope increases. 

The proper dynamical formulation of a Levy flight on the semi-infinite interval with an absorbing boundary condition at x = 0, and thus the determination of the first passage time density, has to ensure that in terms of the random walk picture jumps across the sink are forbidden. This objective can be consistently achieved by setting/(x, t) = 0 on the left semi-axis, i.e., actually removing the particle when it crosses the point x = 0. This procedure formally corresponds to the modified dynamical equation... 

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