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摘要
Liu and Shen discussed the role of stochastic suppression on the explosive solution by a polynomial noise for a deterministic differential system satisfying a general polynomial growth condition. They further showed that the global solution of the corresponding perturbed system grows at most polynomially. However, the estimation of the asymptotic property of polynomial growth is rough, and we see the necessity to develop a more accurate estimation which is the main motivation of the present paper. As to the existence of time delays, we aim to discuss the stochastic roles of the polynomial noise for a deterministic delay differential system with the general polynomial growth condition. We show that a properly chosen polynomial stochastic noise not only can guarantee the existence and uniqueness of the global solution of the stochastically perturbed delay differential system, but also can make almost every sample path of the global solution grow at most with polynomial rate and even decay to the zero solution exponentially.
