Statistical testing-based framework for differentiating anomalous diffusion models with constant and random parameters

Anomalous diffusion describes processes where the mean squared displacement scales non-linearly with time, $E\left(X^2\left(t\right)\right)\sim t^{\beta }$, where $\beta$ is termed the anomalous exponent. This behavior, seen in complex systems like biological cells, often defies standard diffusion models. Classical models such as fractional Brownian motion (FBM) and scaled Brownian motion (SBM) assume constant exponents, failing to capture dynamics with varying anomalous parameters. To address this limitations, models like FBM with random exponents (FBMRE) and SBM with random exponents (SBMRE) were introduced. This work proposes an universal procedure based on statistical testing framework that distinguishes between anomalous diffusion models with constant and random anomalous exponents using time-averaged statistics and their ratio-based counterparts. A novel procedure for optimizing time lag selection via divergence measure (here the Hellinger distance) is also proposed. The introduced methodology applies broadly to constant vs. random anomalous diffusion scenarios, with effectiveness depending on statistic selection, time lags, and process properties, as shown in simulations (with the two-point distribution of anomalous exponent) and real-world data analysis.