  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.
