  Understanding the role of active fluctuations in physics is a problem in
statu nascendi appearing both as a hot topic and a major challenge. The reason
for this is the fact that they are inherently non-equilibrium. This feature
opens a landscape of phenomena yet to be explored that are absent in the
presence of thermal fluctuations alone. Recently a paradoxical effect has been
briefly communicated in which a free particle transport induced by active
fluctuations in the form white Poisson shot noise can be enormously boosted
when the particle is additionally subjected to a periodic potential. In this
work we considerably extend the original predictions and investigate the impact
of statistics of active noise on the occurrence of this effect. We construct a
toy-model of the jump-relaxation process that allow us to identify different
regimes of the free particle transport boost and explain their corresponding
mechanisms. Moreover, we formulate and interpret the conditions for statistics
of active fluctuations that are necessary for the emergence of giant
enhancement of the free particle transport induced by the periodic potential.
Our results are relevant not only for microscopic physical systems but also for
biological ones such as e.g. living cells where fluctuations generated by
metabolic activities are active by default.
