  The weak noise limit of dissipative dynamical systems is often the most
fascinating one. In such a case fluctuations can interact with a rich
complexity frequently hidden in deterministic systems to give rise of
completely new phenomena that are absent for both noiseless and strong
fluctuations regimes. Unfortunately, this limit is also notoriously hard to
approach analytically or numerically. We reinvestigate in this context the
paradigmatic model of nonequlibrium statistical physics consisting of inertial
Brownian particle diffusing in a tilted periodic potential by exploiting the
state of the art computer simulations of unprecedented time scale. In contrast
to the previous results on this long standing problem we draw an inference that
in the parameter regime for which the particle velocity is bistable the
lifetime of ballistic diffusion diverges to infinity when thermal noise
intensity tends to zero, i.e. an everlasting ballistic diffusion emerges. As a
consequence the diffusion coefficient does not reach its stationary constant
value.
