Mean-square asymptotic stability of linear hereditary systems

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A sufficient condition is given for the mean-square asymptotic stability of the trivial solution of time-homogeneous stochastic linear functional differential equations. We use the stochastic Lyapunov function method and the resulting condition is, in some sense, an extension of the condition for linear ordinary differential equations. This condition is especially useful for equations with small noise and small delays and/or small delay terms. Some extensions for the time-inhomogeneous case are also discussed.