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Fractional and scaled Brownian motion on the sphere: The effects of long-time correlations on navigation strategies

We analyze \emph{fractional Brownian motion} and \emph{scaled Brownian motion} on the two-dimensional sphere $\mathbb{S}^{2}$. We find that the intrinsic long time correlations that characterize fractional Brownian motion collude with the specific dynamics (\emph{navigation strategies}) carried out on the surface giving rise to rich transport properties. We focus our study on two classes of navigation strategies: one induced by a specific set of coordinates chosen for $\mathbb{S}^2$ (we have chosen the spherical ones in the present analysis), for which we find that contrary to what occurs in the absence of such long-time correlations, \emph{non-equilibrium stationary distributions} are attained. These results resemble those reported in confined flat spaces in one and two dimensions [Guggenberger {\it et al.} New J. Phys. 21 022002 (2019), Vojta {\it et al.} Phys. Rev. E 102, 032108 (2020)], however in the case analyzed here, there are no boundaries that affects the motion on the sphere. In contrast, when the navigation strategy chosen corresponds to a frame of reference moving with the particle (a Frenet-Serret reference system), then the \emph{equilibrium} \emph{distribution} on the sphere is recovered in the long-time limit. For both navigation strategies, the relaxation times towards the stationary distribution depend on the particular value of the Hurst parameter. We also show that on $\mathbb{S}^{2}$, scaled Brownian motion, distinguished by a time-dependent diffusion coefficient with a power-scaling, is independent of the navigation strategy finding a good agreement between the analytical calculations obtained from the solution of a time-dependent diffusion equation on $\mathbb{S}^{2}$, and the numerical results obtained from our numerical method to generate ensemble of trajectories.