First Passage Times for Variable-Order Time-Fractional Diffusion

Abstract

We derive the asymptotic first passage time (FPT) distribution for space-dependent variable-order time-fractional diffusion, where the fractional exponent $\hat{I}\pm (x)$ varies with position. For any sufficiently smooth $\hat{I}\pm (x)$ on a finite domain with absorbing and reflecting boundaries, we show that the survival probability decays as $\hat{I}¨(t)\sim C{t}^{-\hat{I}{\pm }_{\ast }}/(\ln t{)}^{\hat{I}}½$, where $\hat{I}{\pm }_{\ast }$ is the minimum value of the fractional exponent and $\hat{I}½$ is determined by the location and shape of the minimum. For a constant fractional exponent $\hat{I}½=0$ and this provides a theoretical prediction that can identify spatially heterogeneous anomalous transport in experiments. We validate the theory against exact Laplace-space solutions and Monte Carlo simulations for linear and nonlinear profiles of $\hat{I}\pm (x)$.