First passage times of Lévy flights coexisting with subdiffusion

We investigate both analytically and numerically the first passage time (FPT) problem in one dimension for anomalous diffusion processes in which Lévy flights and subdiffusion coexist. We analyze the FPT for three subclasses of Lévy stable motions: (i) symmetric Lévy motions characterized by Lévy index μ, 0<μ<2, and skewness parameter β=0, (ii) one-sided Lévy motions with μ, 0<μ<1, and skewness β=1, and (iii) two-sided skewed Lévy motions, the extreme case, 1<μ<2, and skewness β=-1. In all three cases the waiting times between successive jumps are heavy tailed with index α. We show that in all three cases the FPT distributions are power laws. Our findings extend earlier studies on FPTs of Lévy flights by considering the interplay between long rests and the Lévy long jumps.