First-passage times for the space-fractional Fokker-Planck equation

We extend the random walk framework to include compounded steps, providing first-passage time (FPT) properties for a class of superdiffusive processes which are governed by the space-fractional spectral Fokker-Planck equation. This first-passage process introduces FPT properties, different from Lévy flights, that account for space-dependent forces and hitting boundaries throughout the path of a jump. The FPT distribution can be derived for different types of barriers and potentials, for which we also provide specific examples. For the one-sided absorbing boundary with no potential on the semi-infinite line, we find that the FPT density scales asymptotically as t^{-1/(2α)-1} for large times, where the parameter α∈(0,1] relates to the power-law behavior for the distribution of the number of compounded steps. This is in agreement with the method of images but different from the Sparre-Andersen scaling t^{-3/2} for corresponding Lévy flights of order 2α. In this case, there exists an optimal space-fractional exponent α to minimize the mean FPT.