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An apparently ideal way to generate continuous bounded stochastic processesis to consider the stochastically perturbed motion of a point of small mass inan infinite potential well, under overdamped approximation. Here, however, weshow that the aforementioned procedure can be fallacious and lead to incorrectresults. We indeed provide a counter-example concerning one of the mostemployed bounded noises, hereafter called Tsallis-Stariolo-Borland (TSB) noise,which admits the well known Tsallis q-statistics as stationary density. Infact, we show that for negative values of the Tsallis parameter q(corresponding to sufficiently large diffusion coefficient of the stochasticforce), the motion resulting from the overdamped approximation is unbounded. Wethen investigate the cause of the failure of Kramers first type approximation,and we formally show that the solutions of the full Newtonian non-approximatedmodel are bounded, following the physical intuition. Finally, we provide a newfamily of bounded noises extending the TSB noise, the boundedness of whosesolutions we formally show.

Boundedness vs Unboundedness of A Noise Linked to Tsallis q-Statistics: The Role of The Overdamped Approximation