Heterogeneous diffusion in an harmonic potential: the role of the interpretation

Diffusion in heterogeneous energy and diffusivity landscapes is widespread in biological systems. However, solving the Langevin equation in such environments introduces ambiguity due to the interpretation parameter $\alpha$, which depends on the underlying physics and can take values in the range $0<\alpha<1$. The typical interpretations are It\^o ($\alpha=0$), Stratonovich ($\alpha=1/2$), and H\"anggi-Klimontovich ($\alpha=1$). Here, we analyse the motion of a particle in an harmonic potential -- modelled as an Ornstein-Uhlenbeck process -- with diffusivity that varies in space. Our focus is on two-phase systems with a discontinuity in environmental properties at $x=0$. We derive the probability density of the particle position for the process, and consider two paradigmatic situations. In the first one, the damping coefficient remains constant, and fluctuation-dissipation relations are not satisfied. In the second one, these relations are enforced, leading to a position-dependent damping coefficient. In both cases, we provide solutions as a function of the interpretation parameter $\alpha$, with particular attention to the It\^o, Stratonovich, and H\"anggi-Klimontovich interpretations, revealing fundamentally different behaviours, in particular with respect to an interface located at the potential minimum.