  We report on novel Brownian, yet non-Gaussian diffusion, in which the mean
square displacement of the particle grows linearly with time, the probability
density for the particle spreading is Gaussian-like, however, the probability
density for its position increments possesses an exponentially decaying tail.
In contrast to recent works in this area, this behaviour is not a consequence
of either a space or time-dependent diffusivity, but is induced by external
non-thermal noise acting on the particle dwelling in a periodic potential. The
existence of the exponential tail in the increment statistics leads to colossal
enhancement of diffusion, surpassing drastically the previously researched
situation known under the label of "giant" diffusion. This colossal diffusion
enhancement crucially impacts a broad spectrum of the first arrival problems,
such as diffusion limited reactions governing transport in living cells.
