Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation

TL;DR

This paper thoroughly analyzes uncoupled continuous-time random walks, solving the master equation for Mittag-Leffler survival probability, exploring their diffusive limits, and clarifying misconceptions in existing literature.

Contribution

It provides an exact solution for the master equation of uncoupled CTRWs with Mittag-Leffler survival probability and discusses their relation to fractional diffusion.

Findings

Solution of the master equation for Mittag-Leffler survival probability

Derivation of the diffusive limit and its connection to fractional diffusion

Clarification of common objections in the literature

Abstract

A detailed study is presented for a large class of uncoupled continuous-time random walks (CTRWs). The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the literature are thoroughly reviewed.