Characterisation of the asymptotic behaviour of scalar linear differential equations with respect to a fading stochastic perturbation

In this paper, we characterise the global stability, boundedness, and unboundedness of solutions of a scalar linear stochastic dierential equation, where the diusion coecient is independent of the state. The dierential equation is a perturbed version of a linear deterministic equation with a globally stable equilibrium at zero. We give conditions on the rate of decay of the noise intensity under which all solutions either tend to the equilibrium, are bounded but tend to zero with probability zero, or are unbounded on the real line. We also show that no other types of asymptotic behaviour are possible.