A Universal Algorithm for Continuous Time Random Walks Limit Distributions

TL;DR

This paper introduces a universal algorithm for computing probability densities of a broad class of Continuous Time Random Walk (CTRW) limit processes, extending previous methods to handle spatially varying waiting times, temporal drifts, and inverse subordinators.

Contribution

It generalizes the DTRW algorithm to virtually all CTRW limit processes under mild conditions, enabling new applications in anomalous diffusion modeling.

Findings

Applicable to CTRWs with spatially varying waiting times

Handles CTRWs with temporal drift affecting short-time behavior

Enables computation of densities for generalized inverse subordinators

Abstract

In this article, we generalize the recent Discrete Time Random Walk (DTRW) algorithm, which was introduced for the computation of probability densities of fractional diffusion. Although it has the same computational complexity and shares the same desirable features (consistency, conservation of mass, strictly non-negative solutions), it applies to virtually every conceivable Continuous Time Random Walk (CTRW) limit process, which we define broadly as the limit of a sequence of jump processes with renewals at every jump. Our only restrictive assumption is the boundedness and continuity of coefficients of the underlying Langevin proceesses. We highlight three main novel use-cases: i) CTRWs with spatially varying waiting times, e.g. for interface problems between two differently anomalous media; ii) (varying) temporal drift, which limits the short-time speed of subdiffusive processes; and iii) the computation of probability densities for generalized inverse subordinators.