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

Dario Domingo,Franco Flandoli,Alberto D’Onofrio

doi:10.1063/1.4977081

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.

Highlights

The influence of extrinsic sources of stochasticity in otherwise deterministic biological systems is very frequently taken into account in an elementary way
The deterministic dynamical system is often perturbed by allowing one or more of its parameters to stochastically fluctuate via a white noise or a colored Gaussian perturbation
The potential U(y) associated with F(y) is such that lim U(y) = +∞. This suggests that an ideal physical “recipe” to generate bounded noises is to consider the overdamped motion of a point in a potential well of infinite height, under the perturbation of a stochastic external “white noise” force

Summary

INTRODUCTION

The influence of extrinsic sources of stochasticity in otherwise deterministic biological systems is very frequently taken into account in an elementary way. The potential U(y) associated with F(y) is such that lim U(y) = +∞ This suggests that an ideal physical “recipe” to generate bounded noises is to consider the overdamped motion of a point in a potential well of infinite height, under the perturbation of a stochastic external “white noise” force. This is a particular limit case of the classical problem of statistical physics studied by Kramers in its hugely influential paper published in 1940.14.

BASIC NOTIONS

Average exit time

CONCLUDING REMARKS