Anomalous diffusion models proliferate because heterogeneous media produce heterogeneous transport. Continuous-time random walks capture waiting-time effects. Fractional Brownian motion captures long-range correlations. Scaled Brownian motion captures time-dependent diffusivity. Each model fits some experimental trajectories but not others, and the field has accumulated frameworks faster than it has connected them.
Lanoiselée, Grebenkov, and Pagnini (arXiv:2603.12775) introduce Randomly Modulated Gaussian Processes as a unifying framework. The construction: take a Gaussian process (which captures displacement correlations) and modulate it with random amplitude fluctuations (which capture environmental heterogeneity). The established models — CTRW, fBM, scaled BM, diffusing diffusivity — emerge as special cases through specific choices of the modulation and correlation structure.
The practical value is diagnostic. The framework derives general expressions for three critical statistical parameters: the non-Gaussian parameter (how far from Gaussian the displacement distribution is), the ergodicity breaking parameter (how much time averages disagree with ensemble averages), and the covariance of squared increments (how fluctuations in step size correlate across time). These three numbers, measurable from a single experimental trajectory, locate the trajectory within the unified space of anomalous diffusion models. Instead of fitting each model separately and comparing likelihoods, the experimentalist measures three statistics and reads off which mechanism dominates.
The structural claim is that the apparent diversity of anomalous diffusion isn't diversity of mechanism but diversity of parameterization within a single mechanism. Displacement correlations and amplitude modulation are the two axes. Every model in the field sits somewhere on that plane.
Lanoiselée, Grebenkov, & Pagnini, “A unifying approach to diffusive transport in heterogeneous media,” arXiv:2603.12775 (March 2026).