In this paper, we study a class of partial differential equations subjected to an additive stationary noise, which is known as a pathwise approximation system for the one driven by white noise. Unlike previous works, we focus on probabilistic properties for the system and its dynamics, which are measured under the \(L^j(\Omega )\) norm, for any \(j\ge 1\) . We first show existence and probabilistic properties of solutions for the approximation system as well as the system driven by white noise. Then we establish for both systems the stochastic inertial manifold structure, which is given in a probabilistic sense rather than the traditional pathwise sense. We also show that solutions and the stochastic inertial manifolds of the approximation system are approaching, under the \(L^j(\Omega )\) norm, to those of the system driven by white noise, respectively. In the end, we illustrate an application of our main results by a simple example.
