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

TL;DR

The paper presents a Markovian, non-homogeneous, non-stationary, and non-ergodic diffusion model based on Generalized Polya Processes, capturing a wide spectrum of anomalous diffusion regimes. By introducing the three-parameter BPM with λ_n(t) = (eta + γ n)/(1 + ρ t) and κ(t) = (1 + ρ t)^{-1}, it shows diffusion speed is governed by the ratio γ/ρ, yielding subdiffusion to hyperballistic behavior while maintaining non-Gaussianity and heavy-tailed waiting times. The work decomposes CLT violations into Moses, Noah, and Joseph effects, deriving exact relations such as M = γ/ρ − 1/2, L = 1/2, J = 1, and H = γ/ρ, and validating these with simulations. It demonstrates the framework’s applicability to epidemics, software reliability, and network traffic, and discusses potential regime transitions via the relaxation function κ(t) for future exploration.

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.

Figures (4)

• Figure 1: Autocorrelation function for the 3p-BPM process for different values of $\gamma/\rho$ in log-scale.
• Figure 2: All different diffusion regimes depicted as a phase diagram plot of $\gamma$ vs $\rho$.
• Figure 3: Displacement distribution (pmf) $X(s + \tau) - X(s)$ of the 3p-BPM model, for ballistic and hyperballistic cases and for different time scales $\tau/s$. The dash-dotted line is the Gaussian density with the same mean and variance. The common mean is marked by the vertical dashed line. The excess kurtosis attains a value of 6 for the ballistic and 9 for the hyperballistic case, being independent of $\tau/s$. For the ballistic we used $\beta = \gamma = \rho = 1$ and for the hyperballistic we used $\beta = \rho = 1$ and $\gamma = 1.5$.
• Figure 4: Ensemble-time averages for a 3p-BPM process with $\gamma/\rho = 3/4$ (superdiffusion).