Scaled Brownian motion as a mean field model for continuous time random walks

We consider scaled Brownian motion (sBm), a random process described by a diffusion equation with explicitly time-dependent diffusion coefficient $D(t) = D_0 t^{α- 1}$ (Batchelor's equation) which, for $α< 1$, is often used for fitting experimental data for subdiffusion of unclear genesis. We show that this process is a close relative of subdiffusive continuous-time random walks and describes the motion of the center of mass of a cloud of independent walkers. It shares with subdiffusive CTRW its non-stationary and non-ergodic properties. The non-ergodicity of sBm does not however go hand in hand with strong difference between its different realizations: its heterogeneity ("ergodicity breaking") parameter tends to zero for long trajectories.

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