Sternberg theorems for random dynamical systems

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Pathwise Random Periodic Solutions of Stochastic Differential Equations (2011, background): “In deterministic and random dynamical systems, to find the existence of stationary solutions and random periodic solutions, and to construct local stable and unstable manifolds near a hyperbolic stationary point is a fundamental problem [1,7,8,11,12,17,22].”
Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients (2006, background): “We would like to point out that there have been extensive works on stability and invariant manifolds of random dynamical systems, and researchers usually assume there is an invariant set (or a single point: a stationary solution or a fixed point, often assumed to be 0), then prove invariant manifold”