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

摘要

In this paper we consider the global stability of solutions of an affinestochastic differential equation. The differential equation is a perturbedversion of a globally stable linear autonomous equation with unique zeroequilibrium where the diffusion coefficient is independent of the state. Wefind necessary and sufficient conditions on the rate of decay of the noiseintensity for the solution of the equation to be globally asymptoticallystable, stable but not asymptotically stable, and unstable, each withprobability one. In the case of stable or bounded solutions, or when solutionsare a.s. unstable asymptotically stable in mean square, it follows that thenorm of the solution has zero liminf, by virtue of the fact that $\|X\|^2$ haszero pathwise average a.s. Sufficient conditions guaranteeing the differenttypes of asymptotic behaviour which are more readily checked are developed. Itis also shown that noise cannot stabilise solutions, and that the results canbe extended in all regards to affine stochastic differential equations withperiodic coefficients.