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The dynamics and invariant manifolds for a nonlocal stochastic Swift-Hohenberg equation with multiplicative noise are investigated. Using a stochastic transformation process, a nonlocal stochastic Swift-Hohenberg equation is studied with either a positive kernel or a non-negative kernel. Then the dynamics, existence, and uniqueness of a global random attractor for the nonlocal stochastic Swift-Hohenberg equation is shown. Moreover, the existence of a local random invariant manifold of the corresponding random dynamical system for the nonlocal stochastic Swift-Hohenberg equation with multiplicative noise is obtained using the technique of a cut-off function and the Lyapunov-Perron method.MSC: 60H15; 76A05; 37H05; 37L55

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