Evaluating Gaussianity of heterogeneous fractional Brownian motion

TL;DR

This paper investigates the non-Gaussian behavior of switching fractional Brownian motion by deriving kurtosis expressions and comparing them with Hellinger distance, providing tools to identify deviations from Gaussianity in heterogeneous anomalous diffusion.

Contribution

It introduces exact kurtosis formulas for SFBM and demonstrates their effectiveness in detecting non-Gaussianity in complex diffusion processes.

Findings

Kurtosis can effectively identify non-Gaussian behavior.

Derived exact kurtosis expressions for SFBM.

Compared kurtosis with Hellinger distance for divergence measurement.

Abstract

Heterogeneous diffusion processes are prevalent in various fields, including the motion of proteins in living cells, the migratory movement of birds and mammals, and finance. These processes are often characterized by time-varying dynamics, where interactions with the environment evolve, and the system undergoes fluctuations in diffusivity. Moreover, in many complex systems anomalous diffusion is observed, where the mean square displacement (MSD) exhibits non-linear scaling with time. Among the models used to describe this phenomenon, fractional Brownian motion (FBM) is a widely applied stochastic process, particularly for systems exhibiting long-range temporal correlations. Although FBM is characterized by Gaussian increments, heterogeneous processes with FBM-like characteristics may deviate from Gaussianity. In this article, we study the non-Gaussian behavior of switching fractional Brownian motion (SFBM), a model in which the diffusivity of the FBM process varies while temporal correlations are maintained. To characterize non-Gaussianity, we evaluate the kurtosis, a common tool used to quantify deviations from the normal distribution. We derive exact expressions for the kurtosis of the considered heterogeneous anomalous diffusion process and investigate how it can identify non-Gaussian behavior. We also compare the kurtosis results with those obtained using the Hellinger distance, a classical measure of divergence between probability density functions. Through both analytical and numerical methods, we demonstrate the potential of kurtosis as a metric for detecting non-Gaussianity in heterogeneous anomalous diffusion processes.