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.
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March 2017
Research Article|
March 01 2017
Boundedness vs unboundedness of a noise linked to Tsallis q-statistics: The role of the overdamped approximation
Dario Domingo;
Dario Domingo
a)
1School of Mathematics,
University of Leeds
, Leeds, United Kingdom
2Department of Mathematics,
University of Pisa
, Pisa, Italy
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Alberto d’Onofrio;
Alberto d’Onofrio
b)
3
International Prevention Research Institute
, Lyon, France
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Franco Flandoli
Franco Flandoli
c)
2Department of Mathematics,
University of Pisa
, Pisa, Italy
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Dario Domingo
1,2,a)
Alberto d’Onofrio
3,b)
Franco Flandoli
2,c)
1School of Mathematics,
University of Leeds
, Leeds, United Kingdom
2Department of Mathematics,
University of Pisa
, Pisa, Italy
3
International Prevention Research Institute
, Lyon, France
J. Math. Phys. 58, 033301 (2017)
Article history
Received:
November 29 2016
Accepted:
February 09 2017
Citation
Dario Domingo, Alberto d’Onofrio, Franco Flandoli; Boundedness vs unboundedness of a noise linked to Tsallis q-statistics: The role of the overdamped approximation. J. Math. Phys. 1 March 2017; 58 (3): 033301. https://doi.org/10.1063/1.4977081
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