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In this paper we characterise the global stability, global boundedness andrecurrence of solutions of a scalar nonlinear stochastic differential equation.The differential equation is a perturbed version of a globally stableautonomous equation with unique equilibrium where the diffusion coefficient isindependent of the state. We give conditions which depend on the rate of decayof the noise intensity under which solutions either (a) tend to the equilibriumalmost surely, (b) are bounded almost surely but tend to zero with probabilityzero, (c) or are recurrent on the real line almost surely. We also show that noother types of asymptotic behaviour are possible. Connections between theconditions which characterise the various classes of long--run behaviour andsimple sufficient conditions are explored, as well as the relationship betweenthe size of fluctuations and the strength of the mean reversion and diffusioncoefficient, in the case when solutions are a.s. bounded.

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