Identifying the interpretation in two-dimensional diffusion processes with power-law spatially dependent diffusivity

Heterogeneous diffusion processes (HDPs) are ubiquitous in macromolecules or submicron tracers in biological cells. The Fokker-Planck equation (FPE) provides a fundamental framework for analyzing these diffusion dynamics. However, its formulations for HDPs with spatially dependent diffusivity are not unique, which can lead to potentially divergent dynamical predictions. We investigate three conventional interpretations (Itô, Stratonovich, and Hänggi-Klimontovich) of FPEs describing two-dimensional diffusion processes with power-law spatially dependent diffusivity. Through analytical solutions of time-dependent probability density functions, which are validated by numerical simulations, we reveal significant interpretation-dependent discrepancies in several physical quantities. These include distinct temporal evolution of mean-square displacement (MSD), survival probabilities, and threshold times at which half of the particles reach a set boundary. Furthermore, we demonstrate that the mean of time-averaged MSDs varies across different interpretations, yielding distinct effective diffusion coefficients. This offers a chance to identify the applicable interpretation by comparing theoretical effective diffusion coefficients with those extracted from single-particle tracking experiments. Our findings reveal that the appropriate interpretation could depend on measurement timescales and system's parameters. We believe that these results are helpful for investigating diffusion dynamics in complex heterogeneous environments.