Evaluating Gaussianity of heterogeneous fractional Brownian motion

TL;DR

The article addresses detecting Gaussianity in heterogeneous diffusion modeled by switching fractional Brownian motion with fluctuating diffusivity. It derives exact kurtosis expressions and analyzes Markovian and non-Markovian switching, corroborated by numerical simulations. The key finding is that kurtosis effectively identifies non-Gaussianity, with Markovian switching converging to Gaussianity at long times while heavy-tailed dwell times sustain non-Gaussian behavior; the Hellinger distance provides a consistent, distribution-based corroboration. The results have practical implications for interpreting heterogeneous diffusion in complex systems, offering a simple, robust diagnostic tool for Gaussianity in data from biophysics and related fields.

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.

Figures (7)

• Figure 1: Evolution of the PDFs of the normalized position in a dichotomous SFBM with exponential dwell times. Each line corresponds to a specific time, with yellow lines representing shorter times and blue lines representing longer times. The plot PDFs correspond to times equal to $t/\Delta = 10, 20, 50, 100, 200, 400,$ and $800$. (a) Subdiffusive case with Hurst exponent $H=0.3$. (b) Superdiffusive case with $H=0.7$. On both panels, the dashed line depicts the standard Gaussian PDF.
• Figure 2: PDFs of the position at time $t=3\Delta$ in a dichotomous SFBM with exponential dwell times (blue line). (a) Subdiffusive case with Hurst exponent $H=0.3$. (b) Superdiffusive case with $H=0.7$. On both panels, the dashed yellow line depicts the mixture of Gaussian PDFs described, while dash-dotted green line corresponds to a Gaussian with variance $2D_- t^{2H}$, and dotted orange line represents a Gaussian with variance $2D_+ t^{2H}$.
• Figure 3: Evolution of the PDF of the normalized position in a dichotomous SFBM with one state having a power-law dwell time and the second state, exponential dwell times. Each line corresponds to a specific time, with yellow lines representing shorter times and blue lines representing longer times. The plot PDFs correspond to times equal to $t/\Delta = 10, 20, 50, 100, 200, 400,$ and $800$. (a) Subdiffusive case with Hurst exponent $H=0.3$. (b) Superdiffusive case with $H=0.7$. On both panels, the dashed line depicts the standard Gaussian PDF.
• Figure 4: Kurtosis for the SFBM with Markovian switching (Exp-Exp) with $H=0.3$. 5th order (dashed yellow line), 10th order (dash-dotted green line), and 100th order (dashed orange line) are shown, where the order describes the number of terms in Eq. (\ref{['eq:kurtosis']}). The empirical kurtosis (purple line) from 100,000 numerical realizations and the correlation time $t_c$ of the dwell times (vertical dashed black line) are also shown. Inset: Detail of small time.
• Figure 5: Empirical kurtosis for two cases. a) $H=0.3$. b) $H=0.7$. The lines are calculated based on $100 \,000$ trajectories. The shaded area corresponds to the 95% confidence interval based on 1 000 samples of length 100 000.