Test free-cumulant connection under non-harmonic confinement for switching diffusion

Determine whether the observed link between long-time cumulants of the position in switching diffusion and the free cumulants of the diffusion-coefficient distribution W(D) persists under confining potentials other than a harmonic trap; specifically, analyze switching diffusion in non-harmonic confinement (e.g., linear potential V(x) = a|x|), derive the stationary distribution in Fourier or real space, and characterize the cumulants to assess the presence or absence of free-cumulant structure.

Background

In Part III, the authors paper switching diffusion—where the diffusion coefficient switches between values drawn from W(D)—and derive exact results for moments, cumulants, and large deviations. Remarkably, they find that at long times (free case) and in the large-β limit under harmonic confinement, cumulants of position are proportional to the free cumulants of W(D).

They pose the question of whether this free-cumulant connection is a special feature of harmonic confinement or a general property that extends to other confining potentials. Establishing this would clarify the robustness of the free-probability structures observed and potentially broaden the applicability of the theory.