  We consider anomalous non-Markovian transport of Brownian particles in
viscoelastic fluid-like media with very large but finite macroscopic viscosity
under the influence of a constant force field F. The viscoelastic properties of
the medium are characterized by a power-law viscoelastic memory kernel which
ultra slow decays in time on the time scale \tau of strong viscoelastic
correlations. The subdiffusive transport regime emerges transiently for t<\tau.
However, the transport becomes asymptotically normal for t>>\tau. It is shown
that even though transiently the mean displacement and the variance both scale
sublinearly, i.e. anomalously slow, in time, <\delta x(t)> ~ F t^\alpha,
<\delta x^2(t)> ~ t^\alpha, 0<\alpha<1, the mean displacement at each instant
of time is nevertheless always larger than one obtained for normal transport in
a purely viscous medium with the same macroscopic viscosity obtained in the
Markovian approximation. This can have profound implications for the
subdiffusive transport in biological cells as the notion of "ultra-slowness"
can be misleading in the context of anomalous diffusion-limited transport and
reaction processes occurring on nano- and mesoscales.
