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

¹  School of Physics and Astronomy, University of Leeds - LS2 9JT Leeds, UK
²  Institute for Theoretical Physics, University of Münster - Wilhelm-Klemm Str. 9, 48149 Münster, Germany

Received: 14 August 2006
Accepted: 7 November 2006

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 Lévy 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.