First-Passage Times for the Space-Fractional Spectral 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 L\'evy 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 $t^{-1/(2\alpha)-1}$ for large times, where the parameter $\alpha \in (0,1]$ 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 to the Sparre-Andersen scaling $t^{-3/2}$ for corresponding L\'evy flights of order $2\alpha$. In this case, there exists an optimal space-fractional exponent $\alpha$ to minimize the mean FPT.