  In this paper we consider the global stability of solutions of an affine
stochastic differential equation. The differential equation is a perturbed
version of a globally stable linear autonomous equation with unique zero
equilibrium where the diffusion coefficient is independent of the state. We
find necessary and sufficient conditions on the rate of decay of the noise
intensity for the solution of the equation to be globally asymptotically
stable, stable but not asymptotically stable, and unstable, each with
probability one. In the case of stable or bounded solutions, or when solutions
are a.s. unstable asymptotically stable in mean square, it follows that the
norm of the solution has zero liminf, by virtue of the fact that $\|X\|^2$ has
zero pathwise average a.s. Sufficient conditions guaranteeing the different
types of asymptotic behaviour which are more readily checked are developed. It
is also shown that noise cannot stabilise solutions, and that the results can
be extended in all regards to affine stochastic differential equations with
periodic coefficients.
