On the Classification of the Asymptotic Behaviour of Solutions of Globally Stable Scalar Differential Equations with Respect to State--Independent Stochastic Perturbations

TL;DR

This paper characterizes the long-term behavior of solutions to a scalar stochastic differential equation with state-independent noise, identifying conditions for stability, boundedness, and recurrence, and establishing the impossibility of other asymptotic behaviors.

Contribution

It provides new conditions based on noise decay rates that determine whether solutions tend to equilibrium, remain bounded, or are recurrent, extending understanding of stochastic stability.

Findings

Solutions tend to equilibrium almost surely under certain noise decay conditions

Solutions are bounded but do not tend to zero under specific parameters

Solutions are recurrent on the real line with probability one

Abstract

In this paper we characterise the global stability, global boundedness and recurrence of solutions of a scalar nonlinear stochastic differential equation. The differential equation is a perturbed version of a globally stable autonomous equation with unique equilibrium where the diffusion coefficient is independent of the state. We give conditions which depend on the rate of decay of the noise intensity under which solutions either (a) tend to the equilibrium almost surely, (b) are bounded almost surely but tend to zero with probability zero, (c) or are recurrent on the real line almost surely. We also show that no other types of asymptotic behaviour are possible. Connections between the conditions which characterise the various classes of long--run behaviour and simple sufficient conditions are explored, as well as the relationship between the size of fluctuations and the strength of the mean reversion and diffusion coefficient, in the case when solutions are a.s. bounded.