Conjugate dynamics on center-manifolds for stochastic partial differential equations

In this paper, we prove that for a random differential equation with the driving noise constructed from a Q-Wiener process and the Wiener shift, there exists a local center, unstable, stable, center-unstable, center - stable manifold, and a local stable foliation, an unstable foliation on the center-unstable manifold, and a stable foliation on the center-stable manifold, the smoothness of which depend on the vector fields of the equation. Also we show that any two arbitrarily local center manifolds constructed as above are conjugate. (C) 2020 Elsevier Inc. All rights reserved.