First Passage Times for Variable-Order Time-Fractional Diffusion

TL;DR

This paper derives the asymptotic distribution of first passage times in space-dependent variable-order time-fractional diffusion, revealing how heterogeneity affects survival probabilities and validating the theory with simulations.

Contribution

It provides a theoretical framework for understanding first passage times in variable-order fractional diffusion with spatial heterogeneity.

Findings

Survival probability decays as t^{- ext{alpha}_*}/( ext{ln} t)^{ u} for large t.

The decay rate depends on the minimum fractional exponent and its spatial profile.

Validation confirms the theoretical predictions against simulations for various alpha(x) profiles.

Abstract

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 $\alpha(x)$.