First-Passage Times for the Space Fractional Fokker-Planck Equation

Abstract: We extend the random walk framework to include compounded steps, providing first-passage time (FPT) properties for a new class of superdiffusive processes, which are governed by the space-fractional spectral Fokker-Planck equation. This first-passage process leads to novel FPT properties, different from Lévy flights and walks, 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 on the semi-infinite line, we find that the FPT density scales as $t^{-1/(2α)-1}$, in agreement with the method of images but in violation of the Sparre-Andersen scaling. In this case, there exists an optimal space-fractional exponent $α$ to minimize the mean FPT.