A fractional diffusion equation for two-point probability distributions of a continuous-time random walk

TL;DR

This paper derives a generalized fractional diffusion equation for two-point probability distributions in subdiffusive continuous-time random walks, extending the single-time equation to better characterize non-Markovian processes.

Contribution

It introduces a closed-form evolution equation for joint two-point distributions, generalizing the fractional diffusion equation to non-Markovian stochastic processes.

Findings

Derived a two-time fractional diffusion equation for CTRWs

Expressed solutions as integral transforms involving inverse Lévy stable processes

Provided explicit formulas for two-time moments of diffusion processes

Abstract

Continuous time random walks are non-Markovian stochastic processes, which are only partly characterized by single-time probability distributions. We derive a closed evolution equation for joint two-point probability density functions of a subdiffusive continuous time random walk, which can be considered as a generalization of the known single-time fractional diffusion equation to two-time probability distributions. The solution of this generalized diffusion equation is given as an integral transformation of the probability distribution of an ordinary diffusion process, where the integral kernel is generated by an inverse L\'evy stable process. Explicit expressions for the two time moments of a diffusion process are given, which could be readily compared with the ones determined from experiments.