  Hydrodynamic memory force or Basset force is known since the 19th-century.
Its influence on Brownian motion remains, however, mostly unexplored. Here, we
investigate its role in nonlinear transport and diffusion within a paradigmatic
model of tilted washboard potential. In this model, a giant enhancement of
driven diffusion over its potential-free limit presents a well-established
paradoxical phenomenon. In the overdamped limit, it occurs at a critical tilt
of vanishing potential barriers. However, for weak damping, it takes place
surprisingly at another critical tilt, where the potential barriers are clearly
expressed. Recently we showed that Basset force could make such a diffusion
enhancement enormously large. In this paper, we discover that even for
moderately strong damping, where the overdamped theory works very well when the
memory effects are negligible, substantial hydrodynamic memory unexpectedly
makes a strong impact. First, the diffusion boost occurs at non-vanishing
potential barriers and can be orders of magnitude larger. Second, transient
anomalous diffusion regimes emerge over many time decades and potential
periods. Third, particles' mobility can also be dramatically enhanced, and a
long transient super-transport regime emerges.
