A note on almost sure exponential stability for stochastic partial functional differential equations.

*(English)*Zbl 0966.60059Summary: In a recent paper, T. Taniguchi [Stochastic Anal. Appl. 16, No. 5, 965–975 (1998; Zbl 0911.60054)] investigated the almost sure exponential stability of the mild solutions of a class of stochastic partial functional differential equations. Precisely, as small delay interval assumption is imposed, sufficient conditions are obtained there to ensure the almost sure exponential stability of the mild solutions of the given stochastic systems. Unfortunately, the main results derived by him are somewhat restrictive to be applied for practical purposes. In the note we shall prove that for a class of stochastic functional differential equations the small delay interval assumption imposed there is actually unnecessary and can be removed.

##### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

35R60 | PDEs with randomness, stochastic partial differential equations |

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\textit{K. Liu} and \textit{A. Truman}, Stat. Probab. Lett. 50, No. 3, 273--278 (2000; Zbl 0966.60059)

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##### References:

[1] | Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052 |

[2] | Hale, J.K., Theory of functional differential equations, (1977), Springer New York · Zbl 0425.34048 |

[3] | Taniguchi, T., Almost sure exponential stability for stochastic partial functional differential equations, Stochastic anal. appl., 16, 5, 965-975, (1998) · Zbl 0911.60054 |

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