Multi-point Distribution Function for the Continuous Time Random Walk

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.

7 pages

References in corpus (8)

• Stochastic Ergodicity Breaking: a Random Walk Approach
• Exact Solution for the Influence of Spectral Diffusion on Single-Molecule Photon-Statistics
• Aging Continuous Time Random Walks
• Aging Correlation Functions for Blinking Nano-Crystals, and Other On - Off Stochastic Processes
• Joint Probability Distributions for a Class of Non-Markovian Processes
• A fractional diffusion equation for two-point probability distributions of a continuous-time random walk
• Non-Poisson dichotomous noise: higher-order correlation functions and aging
• Strong Correlations Between Fluctuations and Response in Aging Transport

Cited by in corpus (14)

• Anomalous transport in the crowded world of biological cells
• Lévy walks
• Nonergodic Subdiffusion from Brownian Motion in an Inhomogeneous Medium
• Continuous Time Random Walks (CTRWs): Simulation of continuous trajectories
• Ergodic properties of continuous-time random walks: finite-size effects and ensemble dependences
• Displacement Autocorrelation Functions for Strong Anomalous Diffusion: A Scaling Form, Universal Behavior, and Corrections to Scaling
• Langevin formulation of a subdiffusive continuous time random walk in physical time
• Transition in the decay rates of stationary distributions of Lévy motion in an energy landscape
• Nonergodicity of $d$-dimensional generalized Lévy walks and their relation to other space-time coupled models
• On the emergence of memory in equilibrium versus non-equilibrium systems
• Generalized autocorrelation function in the family of deterministic and stochastic anomalous diffusion processes
• Ubiquitous Dynamical Time Asymmetry in Measurements on Materials and Biological Systems
• Generalized Khinchin Theorem for a Class of Aging Processes
• Strong overlap of deterministic and stochastic dynamics in a super-diffusive regime