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

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.

Properties of bounded stochastic processes employed in biophysics

“…First, however, a comparison with the TSB case is crucial. In that case, as shown in [28], attaining the boundaries (q < 0) has major consequences on the uniqueness and boundedness of the solution. In the case of the DCL SDE, instead, both the drift and the diffusion of the SDE are well defined at ±1, and the unique solution of the SDE remains bounded for any value of δ and θ (Theorem 4.2), even if the boundaries are attained.…”

“…Questions of existence, uniqueness and boundedness of the solution to the above SDE are strongly related to the particular value of q. In [28] it is shown that uniqueness and boundedness are lost for q < 0, and a physical interpretation of the phenomenon is provided. For q ∈ [0, 1), instead, equation (9) admits a unique strong solution bounded in • I (as shown in Section 3), whose stationary density is given by…”

“…However, by means of the tools and results summarized in Appendix C, we now show that with probability one a solution of the TSB equation does never reach the endpoints of I, which will yield a unique global strong solution of (9). The scale function of the TSB equation (9), as computed in [28], reads as follows:…”

“…• I with probability one, which translates into the fact that the local strong solution is global in time, and it does never leave The case q < 0 has been investigated in detail in [28], where it is shown that the solution X t would in this case reach the endpoints ±1 in finite time, almost surely. Notice that the result of Theorem 3.1 can be extended to any random initial condition X 0 with state space in…”

“…0 < f (δ) < 1, the solution to (81) attains the boundaries with probability one. Following the lines of [28], uniqueness and boundedness are lost. Notice that the same conclusion holds true even more in the case…”