First Passage Times for Variable-Order Time-Fractional Diffusion

We derive the asymptotic first passage time (FPT) distribution for space-dependent variable-order time-fractional diffusion, where the fractional exponent $\alpha(x)$ varies with position. For any sufficiently smooth $\alpha(x)$ on a finite domain with absorbing and reflecting boundaries, we show that the survival probability decays as $\Psi(t)\sim C\,t^{-\alpha_*}/(\ln t)^{\nu}$, where $\alpha_*$ is the minimum value of the fractional exponent and $\nu$ is determined by the location and shape of the minimum. For a constant fractional exponent $\nu=0$ 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 $\alpha(x)$.