Asymmetric space-time correlated continuous-time random walk

In this manuscript, we present an asymmetric space-time correlated continuous-time random walk model. The waiting time distribution is considered to follow a power law omega(t) similar to t(-(1+alpha)) with 0 < alpha < 2, whereas the jump lengths are governed by a pair of asymmetric time-correlated conditional Gaussian-like distributions. The diffusive behaviors are analyzed and discussed by calculating the variance of the displacement analytically and numerically. Results reveal that the space-time correlation and the asymmetry can yield quite nontrivial anomalous diffusive behaviors: for 0 < alpha < 1, the diffusion presents a bi-fractional form, and for 1 < alpha < 2, it displays a multi-fractional one. However, after experiencing a crossover caused by all diffusive terms at intermediate timescales, the diffusion always evolves towards a steady state that is characterized by the term with the largest diffusion exponent at large timescales.