Dynamics of stochastic differential equations with memory driven by colored noise

In this paper, we will show two approaches to analyze the dynamics of a stochastic partial differential equation (PDE) with long time memory, which does not generate a random dynamical system and, consequently, the general theory of random attractors is not applicable. On the one hand, we first approximate the stochastic PDEs by a random one via replacing the white noise by a colored one. The resulting random equation does generate a random dynamical system which possesses a random attractor depending on the covariance parameter of the colored noise. On the other hand, we define a mean random dynamical system via the solution operator and prove the existence and uniqueness of weak pullback mean random attractors when the problem is driven by a more general white noise.