Upper Semicontinuity of Random Attractors for Random Differential Equations with Nonlinear Diffusion Terms I: Finite-Dimensional Case

The upper semicontinuity of random attractors for stochastic/random (partial) differential equations with nonlinear diffusion term is an unsolved problem. In this paper, we first show the existence of random attractor for the random differential equation with nonlinear diffusion term driven by the approximation of the fractional noise, and then prove the upper semicontinuity of the random attractors when the intensity of the approximations tends to zero. The obtained result partly gives an answer to this problem.