Limiting behaviour of smooth inertial manifolds for random retarded differential equations and their Galerkin approximations

Abstract In this paper, we are mainly devoted to the limiting behaviour of smooth inertial manifolds for a class of random retarded differential equations with a singular parameter $\delta$ and their Galerkin approximations, which have not been considered before. Under appropriate conditions, we show not only that the inertial manifolds for this class of random retarded equations converge pointwise to those of the corresponding stochastic equations driven by white noise as $\delta\rightarrow 0$ , but also that the inertial manifolds of their Galerkin approximations converge pointwise to those of stochastic equations driven by white noise described above under the simultaneous limits $\delta\rightarrow 0$ and $M\rightarrow +\infty$ , where $M$ denotes the dimension index of the orthogonal projection operator $P_M$ in the Galerkin scheme.