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

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.

6 pages, 1 figure

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• A Semi-Markov Algorithm for Continuous Time Random Walk Limit Distributions
• Generalized autocorrelation function in the family of deterministic and stochastic anomalous diffusion processes
• Generalized Khinchin Theorem for a Class of Aging Processes
• Strong overlap of deterministic and stochastic dynamics in a super-diffusive regime