  We consider different Markovian embedding schemes of non-Markovian stochastic
processes that are described by generalized Langevin equations (GLE) and obey
thermal detailed balance under equilibrium conditions. At thermal equilibrium
superdiffusive behavior can emerge if the total integral of the memory kernel
vanishes. Such a situation of vanishing static friction is caused by a
super-Ohmic thermal bath. One of the simplest models of ballistic
superdiffusion is determined by a bi-exponential memory kernel that was
proposed by Bao [J.-D. Bao, J. Stat. Phys. 114, 503 (2004)]. We show that this
non-Markovian model has infinitely many different 4-dimensional Markovian
embeddings. Implementing numerically the simplest one, we demonstrate that (i)
the presence of a periodic potential with arbitrarily low barriers changes the
asymptotic large time behavior from free ballistic superdiffusion into normal
diffusion; (ii) an additional biasing force renders the asymptotic dynamics
superdiffusive again. The development of transients that display a
qualitatively different behavior compared to the true large-time asymptotics
presents a general feature of this non-Markovian dynamics. These transients
though may be extremely long. As a consequence, they can be even mistaken as
the true asymptotics. We find that such intermediate asymptotics exhibit a
giant enhancement of superdiffusion in tilted washboard potentials and it is
accompanied by a giant transient superballistic current growing proportional to
$t^{\alpha_{{\rm eff}}}$ with an exponent $\alpha_{\rm eff}$ that can exceed
the ballistic value of two.
