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

Abstract: 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.