Multi-point Distribution Function for the Continuous Time Random Walk

TL;DR

This paper derives an explicit Fourier-Laplace expression for the multi-point distribution function of continuous time random walks, extending previous single-point results, and explores correlations and fractional diffusion equations.

Contribution

It provides a new explicit formula for the multi-point distribution function of CTRWs, generalizing earlier single-point results and linking to fractional diffusion equations.

Findings

Derived explicit Fourier-Laplace transform for multi-point distribution

Established structure as convolution of single-point distributions

Analyzed correlation functions and connection to fractional diffusion

Abstract

We derive an explicit expression for the Fourier-Laplace transform of the two-point distribution function $p(x_1,t_1;x_2,t_2)$ of a continuous time random walk (CTRW), thus generalizing the result of Montroll and Weiss for the single point distribution function $p(x_1,t_1)$. The multi-point distribution function has a structure of a convolution of the Montroll-Weiss CTRW and the aging CTRW single point distribution functions. The correlation function $<x(t_1) x(t_2) >$ for the biased CTRW process is found. The random walk foundation of the multi-time-space fractional diffusion equation [Baule and Friedrich [{\em Europhysics Letters} {\bf 77} 10002 (2007)] is investigated using the unbiased CTRW in the continuum limit.