Boundedness vs unboundedness of a noise linked to Tsallis q-statistics: The role of the overdamped approximation

An apparently ideal way to generate continuous bounded stochastic processes is to
consider the stochastically perturbed motion of a point of small mass in an infinite
potential well, under overdamped approximation. Here, however, we show that the
aforementioned procedure can be fallacious and lead to incorrect results. We indeed
provide a counter-example concerning one of the most employed bounded noises, here-
after called Tsallis-Stariolo-Borland (TSB) noise, which admits the well known Tsallis
q-statistics as stationary density. In fact, we show that for negative values of the Tsallis
parameter q (corresponding to sufficiently large diffusion coefficient of the stochas-
tic force), the motion resulting from the overdamped approximation is unbounded.
We then investigate the cause of the failure of Kramers first type approximation, and
we formally show that the solutions of the full Newtonian non-approximated model
are bounded, following the physical intuition. Finally, we provide a new family of
bounded noises extending the TSB noise, the boundedness of whose solutions we
formally show.