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Classification of the Asymptotic Behaviour of Globally Stable Differential Equations with Respect to State-independent Stochastic Perturbations

In this paper we consider the global stability of solutions of a nonlinear 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. Contingent on a dissipative condition characterising the asymptotic stability of the unperturbed equation, necessary and sufficient conditions on the rate of decay of the noise intensity for the solution of the equation to be a.s. globally asymptotically stable, contingent on some weak and noise independent reversion towards the equilibrium when the solution is far from equilibrium. Under a stronger equilibrium reverting condition, we may classify whether the solution globally asymptotically stable, stable but not asymptotically stable, and unstable, each with probability one purely in terms of the asymptotic intensity of the noise. Sufficient conditions guaranteeing the different types of asymptotic behaviour which are more readily checked are developed.