Hydrodynamic memory force or Basset force has been 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 [Phys. Rev. Lett. 87, 010602 (2001)PRLTAO0031-900710.1103/PhysRevLett.87.010602] 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 [Phys. Rev. Lett. 123, 180603 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.180603] 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 nonvanishing 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 supertransport regime emerges.
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