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Mean-square random attractors of stochastic delay differential equations with random delay. (English) Zbl 1316.34083

The existence of a random attactor is established for a mean-square random dynamical system (MS-RDS) generated by a stochastic delay equation (SDDE) with random delay for which the drift term is dominated by a nondelay component satisfying a one-sided dissipative Lipschitz condition. It is shown by Razumikhin-type techniques that the solution of this SDDE is ultimately bounded in the mean-square sense and that solutions for different initial values converge exponentially together as time increases in the mean-square sense. Consequently, similar boundedness and convergence properties hold for the MS-RDS and imply the existence of a mean-square random attractor for the MS-RDS that consists of a single stochastic process. The main results are of interest to scholars in this field.

MSC:

34K50 Stochastic functional-differential equations
37H10 Generation, random and stochastic difference and differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34K25 Asymptotic theory of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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