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A criterion for stationary solutions of retarded linear equations with additive noise. (English) Zbl 1237.60051

The authors consider stationary (mild) solutions for a class of retarded functional linear differential equations with additive noise in Hilbert spaces. They first introduce a family of Green operators for the stochastic systems and they establish stability results which will play an important role in the investigation of stationary solutions. A criterion imposed on the Green operators is presented to identify a unique stationary solution for the systems considered. Under strong quasi-Feller property, it is shown that this criterion is a sufficient and necessary condition to guarantee a unique stationary solution, based on a method having its origins in optimal control theory.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
35R60 PDEs with randomness, stochastic partial differential equations
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