A non-homogeneous, non-stationary and path-dependent Markov anomalous diffusion model

Abstract

A novel probabilistic framework for modelling anomalous diffusion is presented. The resulting process is Markovian, non-homogeneous, non-stationary, non-ergodic, and state-dependent. The fundamental law governing this process is driven by two opposing forces: one proportional to the current state, representing the intensity of autocorrelation or contagion, and another inversely proportional to the elapsed time, acting as a damping function. The interplay between these forces determines the diffusion regime, characterized by the ratio of their proportionality coefficients. This framework encompasses various regimes, including subdiffusion, Brownian non-Gaussian, superdiffusion, ballistic, and hyperballistic behaviours. The hyperballistic regime emerges when the correlation force dominates over damping, whereas a balance between these mechanisms results in a ballistic regime, which is also stationary. Crucially, non-stationarity is shown to be necessary for regimes other than ballistic. The model's ability to describe hyperballistic phenomena has been demonstrated in applications such as epidemics, software reliability, and network traffic. Furthermore, deviations from Gaussianity are explored and violations of the central limit theorem are highlighted, supported by theoretical analysis and simulations. It will also be shown that the model exhibits a strong autocorrelation structure due to a position dependent jump probability.