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, hereafter 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 stochastic 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.

1.
A.
d’Onofrio
,
Bounded Noises in Physics, Biology, and Engineering
(
Birkäuser
,
2013
).
2.
C.
Tsallis
, in
Nonextensive Statistical Mechanics and Its Applications
, edited by
S.
Abe
and
Y.
Okamoto
(
Springer-Verlag
,
2001
), pp.
3
97
.
3.
D.
Prato
and
C.
Tsallis
,
Phys. Rev. E
60
,
2398
(
1999
).
4.
C.
Tsallis
,
S.
Levy
,
A.
Souza
, and
R.
Maynard
,
Phys. Rev. Lett.
75
,
3589
(
1995
).
5.
7.
H.
Wio
and
R.
Toral
,
Phys. D
193
,
161
(
2004
).
8.
M.
Fuentes
,
C.
Tessone
,
H.
Wio
, and
R.
Toral
,
Fluctuations Noise Lett.
3
,
L365
(
2003
).
9.
M.
Fuentes
,
R.
Toral
, and
H.
Wio
,
Phys. A
295
,
114
(
2001
).
10.
M.
Fuentes
,
H.
Wio
, and
R.
Toral
,
Phys. A
303
,
91
(
2002
).
11.
A.
d’Onofrio
,
Phys. Rev. E
81
,
021923
(
2010
).
12.
A.
Baura
,
M.
Sen
,
G.
Goswami
, and
B.
Bag
,
J. Chem. Phys.
134
,
044126
(
2011
).
13.
H.
Wio
and
R.
Deza
, in
Bounded Noises in Physics, Biology, and Engineering
, edited by
A.
d’Onofrio
(
Birkäuser
,
2013
), pp.
43
58
.
15.
I.
Karatzas
and
S.
Shreve
,
Brownian Motion and Stochastic Calculus
, 2nd ed., SDEs: A Study of the One-Dimensional Case (
Springer
,
1998
), Chap. 5.5.
16.
C.
Gardiner
,
Handbook of Stochastic Methods
, 3rd ed. (
Springer
,
2004
), Chap. 5.2.
17.
P.
Hänggi
and
P.
Jung
,
Adv. Chem. Phys.
89
,
239
(
1995
).
18.
P.
Hänggi
and
P.
Jung
,
Phys. Rev. A
10
,
4464
(
1987
).
19.
R.
Becker
,
Theorie der Wärme
, 3rd ed. (
Springer-Verlag
,
1985
).
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