<i>P</i>-variation statistics of random diffusivity processes

Abstract Anomalous diffusion phenomena, characterized by non-linear mean-squared displacement (MSD), are ubiquitous in a wide range of natural and artificial systems. The methods of distinguishing between different diffusion processes are essential for understanding the underlying mechanisms of the anomalous diffusion phenomena. As one of the typical methods, p -variation statistical analysis has been verified to distinguish the processes with the same MSD, such as fractional Brownian motion (FBM) and scaled Brownian motion. This paper focuses on the random diffusivity processes and develops the p -variation statistics to distinguish between them. We derive the explicit p -variation formulas of two kinds of random diffusivity processes, generalized minimal diffusing diffusivity and generalized gray Brownian motion, together with their variants in the version of FBM. We find that the p -variation formulas of the two kinds of processes are both stochastic processes due to the randomness of diffusivity. The results of the two kinds of processes also show some similarities and differences, which reveals the essential connections between them from the aspect of p -variation statistics. The theoretical p -variation formulas are verified through numerous simulations. We also demonstrate the application of p -variation statistics to the analysis of particles’ trajectories, facilitating the identification of the appropriate random diffusivity processes.