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

TL;DR

This paper demonstrates that the common overdamped approximation can incorrectly suggest unbounded behavior in certain bounded noises linked to Tsallis q-statistics, and introduces a new family of bounded noises.

Contribution

It reveals the limitations of the overdamped approximation for Tsallis-based noises and introduces a new family of genuinely bounded stochastic processes.

Findings

Overdamped approximation can lead to unbounded solutions for certain Tsallis q-parameters.

Full Newtonian models show solutions remain bounded, contrary to overdamped predictions.

A new family of bounded noises extending TSB noise is proposed and validated.

Abstract

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.