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Almost sure exponential stability for stochastic neutral partial functional differential equations. (English) Zbl 1115.60064
A study on a new type of stochastic neutral functional differential equations in a real separable Hilbert space is presented. Some results on the existence and uniqueness of a mild solution and exponential stability of the moments of a solution are given. Also almost sure asymptotic behaviour of the sample paths is investigated. The obtained results are generalization of those reported in [Stochastics Stochastics Rep. 53, No. 1–2, 41–52 (1995; Zbl 0854.60051); Stochastic Anal. Appl. 16, No. 5, 965–975 (1998; Zbl 0911.60054)] and in the paper [“A note on almost sure exponential stability for stochastic partial functional differential equations”, Stat. Probab. Lett. 50, 273–278 (2000; Zbl 0966.60059)] by K. Liu and A. Truman. An example that illustrates the theory is presented as well.

60H20Stochastic integral equations
34K50Stochastic functional-differential equations
35B35Stability of solutions of PDE
35R60PDEs with randomness, stochastic PDE